mtt 


rMAi 


MAT 


THEORY  OF   FUNCTIONS 

OF  A  COMPLEX  VARIABLE 


BY 

DR.    HEINRICH    BURKHARDT 

O.    PROFESSOR,   TECHNICAL   SCHOOL,    MUNICH 


AUTHORIZED  TRANSLATION  FROM  THE  FOURTH  GERMAN  EDITION 

WITH   THE   ADDITION   OF    FIGURES   AND    EXERCISES 
BY 

S.    E.    RASOR,    M.Sc. 

PROFESSOR   OF    MATHEMATICS,   THE   OHIO    STATE   UNIVERSITY 


REVISED 


D.   C.    HEATH    &    CO.,   PUBLISHERS 
BOSTON  NEW  YORK  CHICAGO 


COPYRIGHT,  1913,  BY 
D.  C.  HEATH  &  Co, 

2B8 


POINTED  IN   U.  S.  A. 


TRANSLATOR'S    PREFACE 

FOR  a  number  of  years  there  has  been  a  feeling  among  many 
teachers  of  mathematics  that  students  would  accomplish  more 
if  they  had  an  introductory  treatise  on  the  Theory  of  Functions 
of  a  Complex  Variable  written  in  English  and  adapted  in  other 
ways  to  the  use  of  students  beginning  their  graduate  work. 
Professor  Burkhardt's  book  Einfnhrung  in  die  Theorie  der  ana- 
lytischen  Funktionen  einer  komplexen  Ver'dnderlicheti  has  seemed 
admirably  suited  to  this  purpose  both  in  the  matter  of  material 
and  arrangement.  The  translation  of  this  text  into  English 
was  undertaken  at  the  suggestion  and  encouragement  of  promi 
nent  American  mathematicians,  sinct  there  is  no  book  in  the 
English  language  treating  the  subject  from  the  point  of  view 
adopted  here. 

The  translation  preserves  the  same  arrangement  of  material 
as  the  original  book,  even  to  the  numbering  of  sections  and 
theorems.  Footnotes  not  in  the  original  text  are  always  signed. 
All  of  the  exercises,  which  include  a  number  of  additional  fig 
ures  and  which  follow  various  sections  and  each  of  the  chap 
ters,  are  added  by  the  translator.  It  is  hoped  that  they  will 
prove  of  assistance  not  only  in  illustrating  and  fixing  the  ideas 
contained  in  the  text  but  as  well  in  stimulating  the  reader  to 
independent  attack  and  further  study  in  larger  treatises.  It 
seemed  best  not  to  give  the  sources  from  which  many  of  these 
exercises  were  obtained  —  some  of  them  being  original,  some 
the  results  of  courses  with  the  late  Professor  Maschke  of  the 


iv  TRANSLATOR'S   PREFACE 

University  of  Chicago,  some  furnished  by  Professor  Osgood  of 
Harvard  University,  and  others  seeming  by  this  time  to  have 
become  common  property. 

In  any  case,  the  aim  has  been  to  place  at  the  disposal  of 
students  such  a  book  as  will  be  of  greater  service  in  obtain 
ing  a  knowledge  of  the  fundamental  principles  underlying  the 
theory  of  functions. 

I  wish  to  acknowledge  my  indebtedness  and  gratitude  to 
Professor  N.  J.  Lennes  of  the  University  of  Montana  for  his 
interest  and  help  in  reading  much  of  the  manuscript ;  to  Pro 
fessor  E.  G.  Bill  and  Dr.  F.  M.  Morgan  of  Dartmouth  College 
for  readfng  the  proof-sheets  and  making  use  of  them  in  class- 
work;  and  especially  to  Professor  J.  W.  Young  of  Dartmouth 
College  for  valuable  counsel  and  criticism. 

In  the  second  edition,  minor  corrections  have  been  made  in 
the  text  and  a  few  changes,  rearrangements,  and  minor  additions 
made  in  the  lists  of  examples. 

S.  E.  RASOR. 

THE  OHIO  STATE  UNIVERSITY, 
April,  1920. 


FROM    THE    PREFACE   TO    THE    FIRST    EDITION 

NEARLY  all  *  of  the  numerous  present  German  textbooks  on  the 
theory  of  functions  treat  the  subject  from  a  single  point  of  view  — 
either  that  of  WEIERSTRASS  or  that  of  RIEMANN.  More  recent  French 
and  English  textbooks  (PICARD,  FORSYTH.  HARKXESS  and  MORLEY) 
have  endeavored  to  close  the  gap  between  the  two  methods ;  in  Ger 
many,  too.  lectures  and  scientific  works  have  gradually  sought  to  unify 
the  two  theories.  But  we  are  yet  in  need  of  a  book  of  moderate  extent 
.  .  .  suitable  to  introduce  beginning  students  to  both  methods.  I  ap 
preciated  very  much  the  need  of  such  a  book  as  I  undertook  to  write 
.  .  .  this  introduction  to  the  theory  of  functions.  RIEM ANN'S  geo 
metrical  methods  are  given  a  prominent  place  throughout  the  book ; 
but  at  the  same  time  an  attempt  is  made  to  obtain,  under  suitable 
limitations  of  the  hypotheses,  that  rigor  in  the  demonstrations  which 
can  no  longer  be  dispensed  with  when  once  the  methods  of  WEIER 
STRASS  are  known. 

The  extended  account  of  the  theory  of  functions  from  RIEMANN'S 
standpoint  is  in  reality  a  preparation  for  his  theory  of  integrals  of  alge 
braic  functions.  This  was  entirely  relevant  while  this  theory  was  the 
only  part  of  RIEMANN'S  plans  concerning  the  theory  of  functions  which 
had  been  carried  out.  In  the  meantime  the  linear  differential  equa 
tions  and  the  automorphic  functions  have  come  into  prominence 
through  the  work  of  POINCAR£  and  KLEIN.  In  an  elementary  book 
we  must  consider  this  most  important  change ;  the  conception  of  the 
fundamental  region  must  have  a  prominent  place  in  such  a  book  and 
must  be  fully  explained  in  connection  with  the  simplest  examples,  such 
as  2n  and  ez.  To  make  room  for  this  I  have  omitted  a  part  of  the 
usual  material,  the  first  of  which  is  the  general  analysis  situs  of  the 
RIEMANN'S  surface  with  a  finite  number  of  sheets. 

*  A  possible  exception  is  HARNACK'S  Outlines  of  the  Differential  and  Integral 
Calculus ;  but  this  cannot  be  recommended  to  beginners. 


VI  FROM   THE   PREFACE  TO  THE   FIRST   EDITION 

The  details  in  the  arrangement  of  the  material  may  be  seen  from 
the  table  of  contents  ;  however,  we  mention  the  following  particulars. 

In  the  first  chapter,  I  have  introduced  the  algebra  of  complex  num 
bers  as  an  algebra  of  number-pairs  without  giving  a  general  theory  of 
number  systems  of  two  (or  more)  units ;  I  have  rather  assumed  with 
out  discussion  the  hypotheses  characteristic  of  the  theory  of  "  ordinary 
complex  numbers." 

The  second  chapter  contains  a  detailed  geometrical  theory  of  the 
elementary  rational  functions  of  a  complex  variable  and  the  conformal 
representations  determined  by  them.  The  transition  from  the  plane 
to  the  sphere  by  stereographic  projection  is  also  considered  ;  it  is  used 
at  various  places  in  the  following  chapters.  The  chapter  closes  with 
a  discussion  of  the  symmetric  invariants  of  four  points  as  a  function 
of  their  double  ratio  ;  this  takes  the  place  of  an  example  (in  itself 
unimportant)  of  a  rational  function  of  a  more  general  character. 

The  fourth  chapter  gives  the  theory  of  single-valued  functions  of 
a  complex  argument  essentially  according  to  CAUCHY  and  RIEMANN. 
After  deriving  the  properties  of  such  functions  in  domains  in  which 
they  are  regular,  a  special  discussion  of  the  sine  and  the  cosine  and 
the  exponential  functions  is  added.  Then  follows  the  theory  of 
isolated  singular  points  in  connection  with  LAURENT'S  theorem ; 
FOURIER'S  series  are  studied  at  the  same  time.  The  discussion  of 
MITTAG-LEFFLER'S  theorem  is  limited  to  the  simple  case  for  which 
the  degree  of  the  additional  polynomial  does  not  become  infinite. 
The  chapter  closes  with  the  applications  of  this  theorem  to  singly 
periodic  functions. 

In  the  fifth  chapter,  which  treats  of  many-valued  functions,  I  have 
ventured  a  change  in  the  usual  arrangement  which  may  not  meet  with 
general  approval :  I  have  put  the  logarithm  and  the  infinitely-sheeted 
RIEMANN'S  surface  accompanying  it  first  and  then  used  its  properties 
in  the  investigation  of  even  the  simplest  irrationalities.  It  is  possible 
to  do  this  without  in  any  way  making  use  of  transcendental  functions  ; 
but  to  be  consistent  we  must  then  avoid  the  trigonometric  form  of  a 
complex  number,  which  is  nothing  else  than  introducing  its  logarithm, 
and  prove  the  existence  of  the  n  roots  of  complex  numbers  by  the  fun- 


PREFACE  TO  THE   SECOND   EDITION  vii 

damental  theorem  of  algebra.  This  appeared  to  be  too  cumbersome 
for  an  elementary  book.  Moreover,  the  general  theory  of  algebraic 
functions  is  entirely  omitted  from  this  chapter  and  in  its  place  a  de 
tailed  discussion  of  the  simplest  cases  is  given.  At  the  close  of  the 
chapter  the  properties  of  the  logarithm  are  used  to  obtain  the  repre 
sentation  of  a  transcendental  integral  function  by  means  of  an  infinite 
product  from  the  division  into  partial  fractions  of  its  logarithmic 
derivative. 

I  have  named  the  sixth  and  last  chapter  -  General  Theory  of  Func 
tions."  The  general  conception  of  analytic  continuation,  the  analytic 
function,  the  RIEMANN'S  surface,  the  natural  boundary,  are  first  treated. 
Besides  this  the  chapter  contains  a  discussion  of  the  principle  of 
reflection. 

Statements  as  to  the  authorities  for  the  definitions  and  theorems  are 
omitted.  At  particular  places  as  they  happen  to  occur  in  the  text  I 
have  given  references  to  the  literature  for  such  readers  as  wish  to 
study  any  of  the  questions  further ;  in  this,  original  sources  have  not 
always  been  named  but  where  possible  just  such  references  have  been 
given  as  seem  suitable  for  the  beginner.  .  .  . 

ANSBACH,  March  26,  1897. 


PREFACE  TO    THE    SECOND    EDITION 

SINCE  the  first  two  and  the  last  three  chapters  have  met  with  general 
approval  outside  of  the  strict  disciples  of  WEIERSTRASS.  I  have  no  occa 
sion  to  make  essential  changes  in  these  chapters.  I  have  given  the 
proof  of  CAUCHY'S  fundamental  theorem  in  the  simple  form  made  pos 
sible  by  the  researches  of  PRINGSHEIM.  GOURSAT,  and  MOORE;  this 
necessitated  a  few  other  changes  and  rearrangements.  Besides,  I  hope 
to  have  gained  in  clearness  at  a  few  places  by  minor  additions. 

On  the  contrary,  the  third  chapter  is  entirely  remodeled:  elementary 
things  are  put  in  my  algebraic  analysis  which  has  appeared  in  the 
meantime  as  part  one  of  these  lectures,  while  a  few  other  theorems 
which  were  not  in  their  place  there  but  which  are  needed  here  now 


Viii  PREFACE  TO  THIRD   AND    FOURTH    EDITIONS 

appear  with  proofs.  Since  the  idea  of  the  double  integral  is  no  longer 
required  in  the  proof  of  CAUCHY'S  theorem,  the  space  thus  required  is 
kept  within  moderate  bounds.  .  .  . 

ZURICH,  October  12,  1903. 


PREFACE  TO  THE  THIRD  AND  FOURTH  EDITIONS 

IN  the  third  edition  I  have  further  added  a  few  examples  of  con- 
formal  representation  and  in  connection  with  them  a  discussion  of  the 
cyclometric  functions  of  a  complex  argument.  ...  In  the  fourth  edi 
tion  the  theorem  of  MORERA  (XIII,  §  38)  and  the  proof  of  the  theorem 
of  WEIERSTRASS  (§  50)  based  upon  it  are  added,  besides  many  im 
provements  in  details. 

H.  BURKHARDT. 

MUNICH,  May  11,  1912. 


CONTENTS 

CHAPTER    I 

COMPLEX    NUMBERS   AND   THEIR   GEOMETRICAL 
REPRESENTATION 

SECTION  PAGE 

1.  On  the  General  Arithmetic  of  Real  Numbers          .         .         .         i 

2.  I  ntroduction  of  Number-pairs  ;  their  Addition  and  Subtraction         3 

3.  Multiplication  of  Number-pairs ;   Number-pairs  as  Complex 

Numbers         .........         7 

4.  Geometrical   Representation   of  Complex   Numbers    by   the 

Points  of  the  Plane 12 

5.  Geometrical  Representation  of  Addition  and  Subtraction  of 

Complex  Numbers  .         .         .         .         .         •         .16 

6.  Geometrical    Representation    of  Multiplication   of   Complex 

Numbers        .         .         .         .         .         •         •         •         .18 

7.  Division  of  Complex  Numbers 20 

Examples  .         . 23 

CHAPTER   II 

RATIONAL  FUNCTIONS  OF  A  COMPLEX  VARI 
ABLE  AND  THE  CONFORMAL  REPRESENTA 
TIONS  DETERMINED  BY  THEM 

8.  General  Introduction;    the  Function  s  +  a  and  the  Parallel 

Translation 28 

9.  The  Function  as 30 

10.  The  Linear  Integral  Function  and  the  General  ''Similarity11 

Transformation 32 

Examples  ..........       37 

11.  The  Function  1/2  and  the  Transformation  by  Reciprocal 

Radii -38 

Examples ...       43 

12.  Division  by  Zero  :  Infinite  Value  of  a  Complex  Variable         .       45 


CONTENTS 


13.  Transition  from  the  Plane  to  the  Sphere  by  Stereographic 

Projection 47 

Examples 53 

14.  The  General  Linear  Fractional    Function   and    the    Circle 

Transformation       ........  54 

Examples  ..........  63 

15.  The  Double  Ratio  Invariant  under  the  Linear  Transforma 

tion         65 

Examples 73 

1 6.  Significance  of  the  Linear  Transformation  on  the  Sphere; 

Collineations  of  Space  corresponding  to  it       .         .         •  77 

17.  The  Function  z'2    .........  82 

1 8.  The  Function  w  —  zn,  n  a  Positive  Integer    ....  87 

Examples 93 

19.  Rational  Integral  Functions 98 

Examples  .         .         .         .         .         .         .         .         .  101 

20.  Rational  Fractional  Functions 102 

Examples .         .         .         .         .         .         .         .         .         .104 

21.  Behavior  of  Rational  Functions  at  Infinity     ....  104 

21  a.    The  Function  w  —  \ \(z  -j-  2~l)     ......  106 

22.  A  somewhat  more  complicated  Example  of  an  Automorphic 

Rational  Function 113 

22  a.   Example  of  a  Rational   Integral    Function   which   is    not 

Linear  Automorphic .119 

Miscellaneous  Examples     .         .         .         .         .         .         .125 

CHAPTER  III 

DEFINITIONS   AND  THEOREMS  ON  THE  THEORY 
OF  REAL  VARIABLES   AND  THEIR  FUNCTIONS 

23.  Sets  of  Points  on  a  Straight  Line 127 

Examples  .         .         .         .         .         .         .         .         .         .130 

24.  Application   of  the  previous  Theorems ;    Continuity  on  an 

Interval 131 

Examples 135 

25.  Sets  of  Points  in  the  Plane 135 

Examples  .         .         .         .         .         .         .         .         .         .142 

26.  Continuity  of  Functions  of  two  Real  Variables       .         .  H3 


CONTENTS  xi 

SECTION  PAGE 

27.  Derivatives 148 

28.  Integration 151 

29.  Curvilinear  Integrals 156 

Miscellaneous  Examples 162 

CHAPTER    IV 

SINGLE-VALUED   ANALYTIC   FUNCTIONS    OF   A 
COMPLEX    VARIABLE 

30.  Introduction          .........  167 

30  a.    Limits  of  Convergent  Sequences  of  Complex  Numbers        .  168 

31.  Continuity  of  Rational  Functions  of  a  Complex  Variable        .  170 

32.  Derivative  of  a  Rational  Function  of  a  Complex  Argument    .  174 

33.  Definition  of  Regular  Functions  of  a  Complex  Argument        .  178 

Examples  .         .         .         .         .         .         .         .         .         .180 

34.  Conformal  Representation 182 

Examples 187 

35.  The  Integral  of  a  Regular  Function  of  a  Complex  Argument .  188 

36.  CAUCHY'S  Theorem 195 

Examples  ..........  197 

37.  Development  of  a  Regular  Function  in  a  Power  Series  .         .  199 

38.  Properties  of  Complex  Power  Series 201 

39.  TAYLOR'S,  MACLAURIN'S  Development  in  a  Power  Series  for 

Complex  Variables 206 

Examples 211 

40.  The  Exponential  Function  and  the  Trigonometric  Functions, 

Sine  and  Cosine     .         .         .         .         .         .         .         .216 

Examples  .         .         .         .         .         .         .         •         •         .219 

41.  Periodicity  of  the  Trigonometric  and  the  Exponential  Func 

tions       222 

42.  Conformal  Representations   Determined  by  Singly  Periodic 

Functions 224 

43.  Poles  or  Non-essential  Singular  Points           ....  227 

44.  Behavior  of  a  Function  of  a  Complex  Argument  at  Infinity ; 

the  Fundamental  Theorem  of  Algebra    ....  229 

Examples •  235 

45.  CAUCHY'S  Theorem  on  Residues    ....                  .  236 

Examples 239 


xii  CONTENTS 

SECTION  PAGE 

46.  The  Theorem  concerning  the  Number  of  Zeros  and  of  Poles. 

Second  Proof  of  the  Fundamental  Theorem  of  Algebra  .  240 

47.  LAURENT'S  Series 247 

Examples  .         .         .         .         .         .         .         .         .         .251 

48.  Behavior  of  a  Regular  Function  in  the  Neighborhood  of  a 

Critical  Point          ........  254 

Examples 256 

49.  FOURIER'S  Series 257 

50.  Sums  of  an  Infinite  Number  of  Regular  Functions          .         .  260 

51.  MITTAG-LEFFLER'S  Theorem 262 

52.  Decomposition   of  Singly  Periodic   Functions    into   Partial 

Fractions 268 

53.  General  Theorems  concerning  Singly  Periodic  Functions       .  273 

Miscellaneous  Examples    .......  277 


CHAPTER   V 

MANY-VALUED     ANALYTIC    FUNCTIONS     OF    A 
COMPLEX   VARIABLE 

54.  Preliminary  Investigation  of  the  Change  of  Amplitude  of  a 

Continuously  Changing  Complex  Quantity      .         .         .  284 

55.  The  RIEMANN'S  Surface  of  the  Amplitude      ....  289 

56.  The  Logarithm 294 

Examples  ..........  301 

57.  Conformal  Representation  Determined  by  the  Logarithm       .  304 

Examples 307 

57  a.    The  Function  tan-1 2- 311 

58.  The  Square  Root 316 

59.  The  RIEMANN'S  Surface  for  the  Square  Root         .         .         .  319 

Examples 324 

60.  Connectivity  of  this  Surface 328 

60  a.    Rational  Functions  of  2  and  s  =  Vz    .         .         .         .  331 

61.  Application  of  CAUCHY'S  Theorems  to  Functions  which  are 

single-valued  on  the  RIEMANN'S  Surface  for  Vz    .  .  333 

62.  The  Functions  V(z  —  a) / (z  —  b)  and  V(z  —  a  )(z  —  b)        .  342 
62  a.    Rational  Functions  of  z  and  <r  =  V(z  —  a)  (z  —  b)      .         .  344 


CONTENTS  xiii 

SECTION  PAGE 

62  b.    Integrals  of  Rational  Functions  of  s  and  the  Square  Root 
of  a  Rational   Integral  Function  of  z   of  the    Second 

Degree  .  _._ ._  .         .         .         .351 

62  c.    The  Function  z  —  w  +  /Vi  —  uP        .         .         .         .         .  355 

62  d.   The  Function  sin-1  w          .......  360 

Examples 364 

63.  The  Function  \J~z -3^5 

64.  The  Equation  s2  =  i  —  z* 368 

Examples 373 

65.  Transition   from   MITTAG-LEFFLER'S   Division   into    Partial 

Fractions  to  WEIERSTRASS'S  Development  in  a  Product  373 

Miscellaneous  Examples 377 

CHAPTER   VI 
GENERAL   THEORY   OF   FUNCTIONS 

66.  The  Principle  of  Analytic  Continuation          ....  382 

67.  General  Construction  of  the  RIEMANN'S  Surface  determined 

by  an  Analytic  Function          ......  385 

68.  Singular   Points  and   Natural    Boundaries    of  single-valued 

Functions 389 

Examples 392 

69.  Singular   Points   and   Natural    Boundaries   of    many-valued 

Functions       .         .         .         .         .         .                  .         .  393 

Examples  ..........  397 

70.  Analytic  Functions  of  Analytic  Functions      ....  397 

71.  The  Principle  of  Reflection 399 

72.  Conformal  Representation  of  a  Triangle  bounded  by  Straight 

Lines  upon  a  Half-plane 403 

73.  Generalization  of  the  Principle  of  Reflection;  Reflection  on 

a  Circle 407 

74.  Conformal  Representation  of  a  Triangle  bounded  by  Arcs  of 

Circles  upon  the  Half-plane    ......  409 

Miscellaneous  Examples    .......  414 

INDEX 422 


THEORY   OF   FUNCTIONS   OF   A 
COMPLEX   VARIABLE 

CHAPTER    I 

COMPLEX  NUMBERS  AND  THEIR  GEOMETRICAL 
REPRESENTATION 

§  1.   On  the  General  Arithmetic  of  Real  Numbers 

ELEMENTARY  ARITHMETIC  is  concerned  with  integers  as  its 
primary  elements  or  objects.  It  shows  how  a  third  number 
can  be  found  by  simple  combinations  (addition,  subtraction,  mul 
tiplication,  etc.)  of  any  two  numbers.  It  then  derives  laws  by 
which  the  result  of  a  certain  series  of  combinations  taken  in 
order,  for  example  a(b-\-c),  can  also  be  found  by  another  series 
of  combinations,  in  this  example  by  ab-\-ac.  It  finally  makes 
use  of  these  laws  to  determine  how  a  quantity,  which  is  to  be 
combined  in  definite  ways  with  other  quantities,  must  be  chosen, 
so  that  the  result  of  these  combinations  shall  be  a  value  pre 
viously  determined.  The  proofs  of  these  laws  and  rules  given 
in  arithmetic  are  of  two  entirely  different  kinds  (cf.  A.  A.*  §  i). 
In  deducing  the  fundamental  laws  it  makes  use  of  the  real  sig 
nificance  of  these  objects  (or  numbers)  and  the  operations 
to  which  they  are  to  be  subjected.  Farther  on  this  real  sig 
nificance  is  not  considered,  but  manipulations  are  performed 

*  In  this  way  BURKHARDT's  Algebraic  Analysis  (26.  edition)  will  be  desig 
nated.  It  is  the  first  part  of  Vol.  I  of  BURKHARDT'S  (1908)  Vorlesungen  iiber 
Funktionentheorie. 

I 


2  I.     COMPLEX   NUMBERS 

merely  with  the  symbols  for  the  objects  and  the  operations  on 
the  basis  of  the  doctrines  of  formal  logic  and  those  laws  of 
arithmetic  already  deduced.  This  distinction  is  later  of  much 
importance.  Originally  the  name  number  was  given  only  to 
"positive  integers"  But  the  needs  of  geometry  require  that  still 
other  elements  be  regarded  as  numbers  (negative,  fractional, 
irrational)  and  as  objects  of  the  "general  arithmetic."  For  these 
more  general  numbers  combinatory  operations  are  then  defined. 
They  are  quite  analogous  to  the  operations  with  integers  and 
receive  the  names  applied  to  the  latter  operations.  It  is  shown 
on  the  basis  of  the  definitions  that  these  operations  with  the  more 
general  numbers  satisfy  the  fundamental  laws  mentioned  above  ; 
hence  it  follows  at  once  that  the  derived  theorems  are  also  true 
for  them.  The  earlier  proofs  given  only  for  positive  integers  are 
valid  here  word  for  word,  since  we  no  longer  make  use  of  the 
properties  of  the  objects,  but  rely  only  upon  the  characteristics 
of  the  operations  already  established  (A.  A.  §  i,  §  9). 

That  it  is  permissible  to  introduce  these  more  general  numbers 
is  based  upon  fa&  freedom  of  scientific  thought  to  choose  its  own 
objects  ;  that  it  is  desirable  to  introduce  such  numbers  is  shown 
by  the  result.  The  negative  and  fractional  numbers  represent 
relations  between  objects  of  daily  experience  in  a  broader  sense 
than  can  be  done  by  the  exclusive  use  of  positive  integers.  The 
irrational  numbers  arise  from  the  desire  to  conceive,  for  scientific 
purposes,  as  absolutely  exact  those  laws  of  space  which  enter 
only  approximately  into  our  experience.  This  desire  cannot  be 
satisfied  by  relations  between  integers  alone. 

In  what  follows,  we  shall  suppose  the  negative  and  the  frac 
tional  numbers  to  be  introduced ;  on  the  contrary,  the  irrational 
number  will  be  used  at  only  a  few  places  in  the  first  two 
chapters. 


§  2.     ADDITION   AND   SUBTRACTION   OF  NUMBER-PAIRS      3 

§  2.   Introduction  of  Number-pairs ;  their  Addition  and  Subtraction 

From  the  algebra  of  the  simple  number  we  pass  next  to  an  al 
gebra  of  the  number-pair  —  the  so-called  "double  algebra."  It 
deduces  a  new  number  from  two  number-pairs  and  seeks  the 
laws  which  underlie  these  combinations  of  number-pairs.  The 
algebra  of  simple  numbers  finds  its  counterpart  in  the  geometry 
of  one-dimensional  configurations  in  so  far  as  it  is  possible  to 
assign  a  definite  number  to  each  point  of  such  configurations,  a 
straight  line  for  example,  and  one  definite  point  to  each  number.* 
We  shall  see  that  the  double  algebra  is  represented  geometri 
cally  by  the  relations  between  the  points  of  two-dimensional  con 
figurations  or  surfaces,  and  in  particular  upon  the  simplest  of 
these,  the  plane  and  the  sphere. 

What  kind  of  combinations  of  number-pairs  we  are  to  consider 
is  arbitrary  with  us;  whether  the  choice  we  make  is  adapted  to 
our  purposes  is  a  question  which  can  be  answered  in  the  affirma 
tive  when  and  only  when  results  have  been  obtained  which  could 
not  be  obtained  at  all  by  other  methods,  or  at  least  not  so  easily. 
But  we  have  two  requirements  to  govern  our  choice.  We  shall 
seek  first  those  combinations  which  obey  the  same  or  nearly 
the  same  laws  as  those  of  simple  numbers  ;  and  besides,  we  shall 
always  keep  in  mind  the  relations  to  geometrical  configurations. 
We  shall  not  raise  the  question  as  to  what  might  be  the  most 
general  combinations  which  satisfy  the  first  requirement  and 
are  also  adapted  to  the  second  ;  but  we  shall  begin  with  the  defi 
nition  of  the  combinations  to  be  considered  and  then  prove  that 
they  obey  the  above  laws  and  show  how  they  are  represented 
geometrically.  In  this  manner  we  follow  the  historical  de 
velopment.  Number-pairs  first  appear  in  the  form  of  "  im- 

*  Known  as  the  CANTOR-DEDEKIND  axiom ;  cf.  PlERPONT,  The  Theory  of 
Functions  of  Real  Variables,  Vol.  I,  p.  79,  —  S.  E.  R. 


4  I.     COMPLEX   NUMBERS 

aginary"  numbers  a  +  b~\/  —  i  in  the  solution  of  algebraic  equa 
tions  of  the  second,  third,  and  fourth  degrees.  Any  hesitation 
on  first  using  these  "  imaginary"  numbers  was  easily  overcome 
by  being  able  to  operate  with  them  as  with  real  numbers,  even 
without  justifying  the  process  or  without  knowing  what  such 
an  imaginary  symbol  in  general  signified.  The  following  defini 
tions  are  all  given  with  the  understanding  that  we  operate  with 
the  imaginary  number  a  -f  bi  as  with  a  real  binomial  and  reduce 
higher  powers  of  i  to  the  first  power  by  the  relation  z  2  +  i  =  o. 

To  operate  with  number-pairs  it  is  necessary  to  define  when 
two  number-pairs  are  equal  to  each  other.     Definition  : 

I.  Two  number-pairs  (a,  b]  and  (c,  d)  are  equal  to  each  other 
when  and  only  when 

a  =  c  and  b  =  d 

(but  not  when  a  =  d  and  b  =  c).  One  equation  between  number- 
pairs  therefore  represents  two  equations  between  simple  numbers. 
The  concepts  "  larger  "  and  "  smaller  "  are  not  immediately 
applicable  to  number-pairs. 

II.  From  the  two  number-pairs  (a,  b)  and  (c,  d)  a  third  number- 
pair 


can  be  formed  in  the  simplest  manner.  A  name  and  a  symbol 
are  needed  for  this  operation.  We  shall  not  introduce  new  ones, 
but  we  shall  borrow  the  name  addition  with  its  symbol  +  from 
the  algebra  of  simple  numbers.  Accordingly  the  third  number- 
pair  is  called  the  sum  of  the  other  two  and  is  written  : 

(i)  (a,b)  +  (c,<I)  =  (a  +  c,t  +  d'). 

A  new  meaning  is  thus  attached  to  these  terms  and  symbols 
(addition,  sum,  -f,  =). 

This   combination  of   number-pairs  is   a  definite  operation, 


§  2.     ADDITION   AND   SUBTRACTION   OF   NUMBER-PAIRS      5 

possible  and  unique   in  ever}'-  case.     Moreover,  this  operation 
obeys  the  commutative  law  (A.  A.  IV,  §  2)  : 


(2) 

and  the  associative  law  (A.  A.  Ill,  §  2)  : 

(3)  [( 


The  first  of  these  laws  is  proved  by  applying  definition  (II) 
to  the  operations  indicated  by  each  side  of  equation  (2).  The 
resulting  number-pairs  (a  +  c,  b  +  d)  and  (c  +  a,  </+  £)  are  equal 
to  each  other  according  to  definition  (I),  since  a  +  c=-c+  a  and 
b  +  d=d+b  according  to  the  commutative  law  for  the  addition 
of  simple  numbers.  Equation  (3)  is  proved  in  a  similar  manner. 

The  further  theorems  in  elementary  algebra  about  the  rear 
rangement  of  the  terms  in  a  sum  of  three  or  more  summands, 
can  be  proved  by  purely  logical  deduction  from  the  commutative 
and  associative  laws  without  returning  to  the  fundamental  mean 
ing  of  the  operation  of  addition.  It  therefore  follows  that  these 
extended  theorems  are  valid  for  the  operations  with  number- 
pairs  just  as  for  ordinary  numbers  (cf..  the  general  remarks  of 
§  i).  Hence  we  may  state  the  following  general  theorem  of 
which  equations  (2)  and  (3)  are  special  cases  : 

III.  /;/  a  sum  oj  any  number  of  number-pairs,  the  separate  sum 
mands  may  be  combi?ied  into  a  smaller  number  of  other  summands 
by  an  arbitrary  selection  and  arrangement. 

We  define  further  : 

IV.  The  number-pair  (—a,  —  b)   is  called  the  opposite  of  tht 
number-pair  (a,  b\ 

V.  The   difference   of  two   number-pairs   is   that  number-pair 
which,  when  added  to  the  subtrahend,  gives  the  minuend. 

From  this  definition,  the  definition  of  sum  (II),  and  the  prop 
erties  of  addition  and  subtraction  of  simple  numbers  we  obtain 


6  I.     COMPLEX   NUMBERS 

Theorem  VI,  which  is  expressed  by  the  equation  : 

(4)  (a,b)-(c,d)  =  (a-c,b-d)-, 

also  Theorem  VII  :  Subtraction  of  a  number-pair  is  the  same  as 
addition  of  the  opposite  number-pair  ',  and  hence  is  a  definite  opera 
tion,  possible  and  unique  in  every  case. 

In  elementary  algebra  the  sum  of  (m)  (a  positive  integer) 
equal  summands  a 

123  m-l      m 

a  +  a  +  a  +  •••  a-\-  a 

is  called  "  the  product  of  the  number  a  by  the  positive  integer  m" 
This  definition  has  a  definite  meaning  in  consequence  of  Theorem 
III  when  applied  to  number-pairs  ;  we  say  : 

VIII.  The  product  of  an  integer  m  and  a  number-pair  (a,  b}  is 
the  sum  of  m  equal  number-pairs  (a,  ^). 

If  we  form  this  sum  according  to  definition  II  and  Theorem 
III  we  obtain  a 

Theorem  IX,  which  is  expressed  by  the  equation  : 

(5)  m(a,  b}  =  (ma,  mb). 

X.  In  case  m  is  a  negative  number,  equation  (5)  is  the  definition 
of  the  product  of  m  and  the  number-pair  (a,  fr). 

XI.  Division  of  a  number-pair  by  an  integer  is  defined  as  the  in 
verse  of  multiplication.      Therefore,    in   consequence   of   equa 
tion  (5): 

(6) 


m  m     m 

XII.    Multiplication  of  a  number-pair  by  a  fraction  is  defined 
as,  "  Multiplication  by  the  numerator  and  division  by  the  denomina 
tor"  (A.  A.  §  15);  and  thus,  from  the  equations  (5)  and  (6),  it 
follows  that  : 
f  \  m,      ,\     (m       m 

(7)  -0,  *)=[-*,  - 

n  n       n 


§  3.     MULTIPLICATION   OF  NUMBER-PAIRS  7 

XIII.    On  the  basis  of  the  definitions  II  and  XII,  every  number- 
pair  can  be  represented  in  the  form 

(8)  (<z,  £)  =**•!  +  &?2 

as  the  algebraic  sum  of  multiples  of  the  two  special  number-pairs  : 


which  are  accordingly  named  UNITS. 

§  3.   Multiplication  of  Number-pairs  ;  Number-pairs  as  Complex 

Numbers 

In  elementary  arithmetic  multiplication  is  considered,  after 
addition,  as  a  second  kind  of  combination  of  two  numbers  to 
produce  a  third  number.  Among  the  properties  characteristic 
of  this  operation  are  the  commutative  law  (A.  A.  Ill,  §  4)  ex 
pressed  by  the  equality 

(  i  )  ab  =  ba, 

and  its  distributive  relation  with  respect  to  addition  (A.  A.  VI, 
§  4)  expressed  by  the  equality 

(2)  a(b  +  c)=-ab-\-ac. 

We  now  inquire  whether  there  exists  also  a  combination  of 
number-/rt7>.r  which  obeys  both  of  these  laws,  that  is,  which  is  in 
itself  commutative  and  which  obeys  the  distributive  law  for 
addition  of  number-pairs  stated  in  §  2.  If  such  a  combination 
exists,  we  name  it  multiplication  and  designate  it  by  juxtaposition 
of  the  factors,  with  or  without  the  point  to  connect  them. 

In  accordance  with  the  representation  of  number-pairs  given 
by  equation  (8),  §  2  and  in  accordance  with  the  requirements 
of  the  commutative  and  the  associative  laws,  the  result  of  the 


8  I.     COMPLEX   NUMBERS 

multiplication  of  any  two  number-pairs  is  determined  when  we 
have  determined  the  product  of  the  two  units  each  by  itself  and 
each  by  the  other  (the  result  being,  of  course,  a  number-pair). 
We  shall  not  attempt  to  answer  the  questions  as  to  what  are  the 
most  general  assumptions  we  are  here  able  to  make  consistent 
with  the  above  hypotheses  or  which  of  these  different  assump 
tions  lead  to  essentially  different  "double  algebras";  but  we 
shall  introduce  directly  those  hypotheses  which  characterize  the 
theory  of  the  so-called  "  ordinary  complex  numbers." 

I.  The  products  of  the  units  are  accordingly  defined  by  the  equa 
tions  : 

(3)  (i,o).(i,o)  =  (i,o), 

(4)  (o,  i)  •  (i,  o)  =  (i,  o)  .  (o,  i)  =  (o,  i), 

(5)  (o,  i)«(o,  i)  =  (-i,o). 

The  meanings  of  these  equations  must  be  made  clear. 

From  equation  (3)  and  from  the  results  of  the  previous  para 
graphs  it  follows  that  all  computation  with  number-pairs,  whose 
second  elements  are  equal  to  o,  is  to  be  performed  just  as  if  the 
second  elements  were  entirely  absent  and  that  therefore  we  are 
to  operate  only  with  the  first  elements  just  as  with  simple  num 
bers.  Equation  (4)  and  the  distributive  law  tell  us  that  the 
number-pair  (a,  o)  is  to  be  treated  as  a  simple  number  when 
multiplying  it  by  another  number-pair.  We  identify  these 
special  number-pairs  directly  with  simple  numbers  as  follows : 

II.  Since  we  may  put 

(6)  (i,°)  =  i, 

it  follows  according  to  equation  (5),  §  2,  that  in  general 

(7)  (",  0)=«- 


§  3.     MULTIPLICATION   OF   NUMBER-PAIRS  9 

Finally,  equation  (5),  on  account  of  equation  (7),  can  be 
written 

(8)  (o,  i)-(o,  i)=-i. 

Accordingly,  it  follows  that : 

III.  While  there  does  not  exist  a  simple   number  which  mul 
tiplied  by  itself  gives  —  I,  there  is  a  number-pair,  viz.  (o,  1)  which 
has  this  property. 

The  problem,  to  find  a  number  which  multiplied  by  itself 
produces  a  given  number,  is  called  in  elementary  algebra 
"  Extraction  of  the  square  root "  and  is  designated  by  ->/ 
(A.  A.  §  46).  If  we  apply  this  symbol  (provisionally  without 
further  discussion)  to  operations  with  number-pairs  we  can 
formulate  Theorem  III  as  follows: 

IV.  The  operation  i?idicated  by  V—  /  is  impossible  in  the  field 
of  simple  numbers,  but  has  (o,  i)  for  a  solution  with  number-pairs. 

(Whether   there    are   other   number-pairs  which    satisfy  this 
operation,  remains  temporarily  undecided;  but  cf.  §  58.) 
Moreover,  we  shall  put 

(9)  (o,  i)=V^7  =  /, 

thus  using   the   symbol  generally  accepted   since   the   time  of 
GAUSS. 

V.  Therefore  according  to  equation  (8) ,  §  2,  every  number-pair 
can  be  written  in  the  form 

(10)  (at&)  =  a  +  bi. 

We  shall  henceforth  use  a  different  terminology  as  follows : 

VI.  What  was   heretofore   named    merely  "a   number'1''    will 
hereafter  be  called  "  a   real  number "  /    and  what  we  heretofore 
called  a  number-pair  will  henceforth  be  named  "a  number"   or 
where  a  more  explicit  statement  is  desired,  "  a  complex  number  " 
(a  complex  quantity). 


IO  I.     COMPLEX   NUMBERS 

We  therefore  enlarge  the  custom  of  representing  an  indeter 
minate  number  by  a  letter,  by  designating  an  arbitrary  complex 
number  by  a  letter  when  the  limitation  to  real  numbers  (or  any 
other  limitation)  is  not  expressly  stated  or  is  not  evident  from 
the  context. 

VII.  In  the  complex  number  a-\-bi,  a  is  called  the  real  part  and 
bi  the  imaginary  part.     A  complex  number  whose  real  part  is  o 
is  called  a  pure  imaginary  number. 

One  must  not  be  led  to  a  wrong  conception  by  this  name ;  as 
we  shall  soon  see,  complex  numbers  are  very  well  suited  to 
represent  definite  relations  between  real  objects. 

The  symbols  thus  introduced  will  be  used  at  once  to  state 
explicitly  the  following  result: 

VIII.  For  the  multiplication  of  any  two  complex  numbers  we 
have,  by  applying  formulas  (i)  to  (5) : 

(i  i)  (a  -f  bi)  (c  +  di)  =  ac  —  bd  +  i(ad+  be). 

The  multiplication  of  complex  numbers  defined  by  this  equa 
tion  is  thus  possible  in  every  case  and  the  result  is  unique. 
That  the  methods  for  the  multiplication  of  real  numbers  hold 
for  these  complex  numbers  is  not  self-evident,  but  must  be 
proved  (just  as  we  proved  the  corresponding  property  for  addi 
tion).  In  this  it  is  sufficient  to  prove  that  the  fundamental  laws 
are  valid  permanently,  in  order  to  show  that  the  laws  derived 
from  them  remain  valid ;  this  has  been  so  completely  discussed 
in  A.  A.  §§  i,  9  as  well  as  here,  §§  i,  2,  that  further  discussion 
is  not  now  necessary.  Multiplication  possesses  three  such  fun 
damental  laws,  viz.,  the  two  stated  at  the  beginning  of  this 
paragraph  and  the  following: 

(12)  (ab)c=a(bc), 

which  is  called  the  associative  law  (A.  A.  IV,  §  4).     To  verify 

the  fact  that  these  three  laws  hold  also  for  the  multiplication  of 


§  3.     MULTIPLICATION   OF  NUMBER-PAIRS  I  I 

complex  numbers  as  defined  by  equation  (n),  it  is  only  neces 
sary  to  carry  out  the  indicated  operations  ;  this  is  left  to  the 
reader,  and  we  state  at  once  the  theorem : 

IX.  The  three  laws  (/),  (2),  (12),  as  well  as  all  the  laws  de 
duced  from  them,  hold  for  multiplication  as  defined  by  equation  (//). 

All  deductions  from  equations  are  naturally  again  equations. 
But  there  is  another  important  property  of  multiplication  of  real 
numbers  which  is  not  expressed  by  an  equation,  but  by  an  in 
equality,  and  on  this  account  cannot  be  deduced  from  the  above 
three  fundamental  laws  alone.  We  refer  to  the  theorem  that 
a  product  cannot  be  zero  unless  one  of  the  factors  is  zero 
(A.  A.  §  13).  We  must  then  show  in  particular  that  this  same 
theorem  holds  for  complex  numbers.  If  the  right  side  of  equa 
tion  (n)  is  to  be  equal  to  zero,  then,  according  to  the  definition 
of  equality  of  two  complex  numbers  given  in  I,  §  2,  we  must 

have 

ac—M—O  c 

It  follows  from  these  equations  by  multiplying  by  the  adjoining 
factors  that 

<'3>  *(^+^=o! 

The  equation  (c--\-d^)  =  o  can  be  satisfied  by  real  values  of 
c,  d,  only  when  c=  o  and  d=o.  But  if  r2  -f  d-=£  o  then  it  fol 
lows  from  equations  (13)  that  a  must  =  o  and  b  must  =  o,  since 
the  theorem  just  cited  holds  for  real  numbers.  If  therefore 


either  a  -f  bi  must  =  o  or  c-\-  di  must  =  o,  that  is,  the  following 
theorem  holds  for  complex  numbers  : 

X.    A  product  cannot  be  zero  unless  one  of  the  factors  is  zero. 


12  I.     COMPLEX  NUiMBERS 

§  4.  Geometrical  Representation  of  Complex  Numbers  by  the 
Points  of  the  Plane 

At  the  beginning  of  our  investigation  (in  §  2)  we  recalled  a 
one-to-one  correspondence  between  the  totality  of  real  numbers 
and  the  totality  of  points  of  a  straight  line,  that  is,  an  arrange 
ment  such  that  to  each  point  there  corresponds  a  definite  num 
ber  (the  "abscissa"  of  the  point)  and  to  each  number  there 
corresponds  a  definite  point.  We  have  likewise  pointed  out 
that  number-pairs  can  be  arranged  in  correspondence  with  the 
points  of  a  surface  as  a  two-dimensional  configuration.  The 
simplest  arrangement  of  this  kind  for  the  points  of  the  plane 
is  the  following  one  due  to  DESCARTES  :  let  us  draw  through  a 
given  point,  the  "  origin "  of  coordinates,  two  straight  lines, 
"  the  x-  and  the  jy-axis,"  perpendicular  to  each  other  ;  drop  per 
pendiculars  from  any  point  of  the  plane  on  these  axes,  and 
designate  the  lengths  cut  off  on  the  axes  by  these  perpendicu 
lars,  measured  from  the  origin  and  taken  with  the  proper  sign,* 
as  the  "  coordinates  "  x,  y  of  the  given  point  (Fig.  i).  In  this 
representation  we  need  only  to  replace  the  number-pair  (x,  y) 
by  the  complex  number  x  +  iy;  we  have  thus  the  relation  of 
the  complex  numbers  to  the  points  of  the  plane  due  to  GAUSS 
and  ARGAND  : 

I.  We  associate  with  each  complex  number  x  -\-  iy  that  point  of 
the  plane  which  has,  in  reference  to  a  fixed  rectangular  system,  the 
coordinates  x,  y  ;  and  conversely,  to  each  point  with  the  coordinates 
x,  y  we  associate  the  complex  number  x  +  iy> 

*  In  general  we  shall  take  the  positive  .r-axis  to  the  right  and  the  positives-axis 
in  front  of  the  observer.  But  in  any  case  let  us  think  of  the  positive  directions  of 
the  axes  as  so  chosen  that  they  can  be  brought  into  the  position  just  indicated  by 
mere  turning  in  the  plane  without  reflection.  Whether  we  use  this  or  the  opposite 
arrangement  is  entirely  immaterial.  However,  it  is  often  convenient  for  the  form 
of  certain  expressions  to  use  a  particular  arrangement  and  at  times,  when  the  desig 
nation  of  signs  is  important,  to  use  the  usual  arrangement.  (Cf.  A.  A.  §§  n,  14.) 


4.     REPRESENTATION   BY   POINTS   OF  THE   PLANE       13 


1 


FIG.  i 


In  this  way  there  corresponds  to  each  complex  number  one 
and  only  one  point  of  the  plane  ;  and  conversely,  to  each  point 
of  the  plane  there  corresponds  one  and  only  one  complex  num 
ber.  Accordingly,  to  each  definite  relation  between  points  of 
the  plane  there  must  be  a 
corresponding  definite  re 
lation  between  complex 
numbers,  and  conversely. 
From  each  theorem  about 
complex  numbers  there 
follows  a  geometrical 
theorem  about  points 
of  the  plane ;  and  con 
versely,  to  each  geo 
metrical  theorem  concern 
ing  points  of  the  plane 

there  is  a  corresponding  theorem  about  complex  numbers.  Of 
course,  every  such  theorem  must  be  proved  "  purely  "  by  methods 
which  belong  only  to  each  particular  case ;  but  powerful  aids  to 
investigation  are  furnished  us  by  the  application  of  known  geo 
metrical  theorems  to  our  function  theory.  This  procedure  is 
justifiable  from  the  standpoint  of  rigor,  whenever  we  are  certain 
of  the  one-to-one  correspondence  between  the  objects  analytically 
related  and  the  geometrical  picture,  and  whenever  we  use  only 
rigorous  geometrical  theorems. 

In  particular,  the  points  of  the  ^c-axis  correspond  to  the  real 
numbers  (VI,  §  3).  It  will  therefore  be  called  the  axis  of  real 
numbers ;  to  the  pure  imaginary  numbers  (VII,  §  3)  correspond 
the  points  of  the  j-axis  (axis  of  pure  imaginary  numbers)* 

From  the  rectangular  coordinates  of  a  point  we  obtain  its  well- 
known  polar  coordinates,  radius  vector  r  and  polar  angle  <£,  by 
*  Also  called  "  real  axis"  and  "  imaginary  axis." 


14  I.     COMPLEX   NUMBERS 

the  equations  (cf.  Fig.  i): 


whose  solution  is  : 
(2) 


The  formulas  (i)  are  also  correct  in  sign  if  the  positive  direction 
of  the  angle  be  so  chosen  that  the  positive  7-axis  makes  an 

angle  of  +  -  with  the  positive  je-axis  *  and  if  r  is  always  taken 

2 

as  positive.  The  foundation  for  these  statements  from  the  theory 
of  trigonometric  functions  of  a  real  angle  (A.  A.  §  76)  is  sup 
posed  to  be  known  here. 

II.  On  the  basis  of  equations  (/)  every  complex  number  can  be 
written  in  the  form  : 

(3)  z  =  x  -\-  iy  =  r  (cos  <£  -f-  i  sin  <£). 

III.  Here  r  is  the  positive  square  root 


/    is    called  the   ABSOLUTE   VALUE  f    of  the  complex   number 
=  x  +  iy  and  is  designated  by 


The  square  of  the  absolute  value  is  called  the  NORM. 

The  absolute  value  of  a  positive  real  number  is  the  number 
itself.  The  absolute  value  of  a  negative  real  number  is  the 
same  number  with  the  opposite  sign  (A.  A.  §  10). 

*  And  thus  with  the  usual  arrangement  for  positive  direction  of  axes,  "  counter 
clockwise  "  (A.  A.  $  n). 

f  Also  called  "  modulus  ";  but  this  word  has  also  other  meanings. 


§  4.    REPRESENTATION   BY   POINTS   OF  THE   PLANE        1 5 

IV.  There  is  no  definite  name  for  the  angle  <£.     \Ve  find  it 
designated  as  argument,  declination,  arcus,  anomalie,  amplitude* 
We  shall  use  the  last-named  term. 

V.  The  factor  cos  $  +  /  sin  <£ 

(direction  factor  of  the  complex  number)  has  the  property  that 
its  absolute  value  =  i . 

VI.  All  the  points   corresponding  to  the  numbers  of  absolute 
value  I  lie  upon  the  unit  circle,  that  is,  upon  the  circle  whose  center 
is  at  the  origin  and  whose  radius  is  unity. 

Whether  it  is  possible  to  put  a  complex  number  in  the  form 

(3)  in  only  one  way  is  quite  an  essential  question.     We  notice 
in  this  connection  that : 

VII.  The  absolute  value  r  is  u?iiquely  defined,  but  there  is  an 
infinite  number  of  values  of  <£  (as  shown  in  goniometry)  which 
satisfy  the  conditions;  all  these  values  can  be  obtai tied  from  any 
one  of  them  by  the  addition  and  subtraction  of  arbitrary  integral 
multiples  of  2  TT  (A.  A.  §  76).     We  must  continually  give  atten 
tion  to  this  many-valued  character  of  the  amplitude  when  using 
complex  numbers  in  the  form  (3).     It  will  be  discussed  again 
more  completely  in  §54. 

VIII.  The  number 

(4)  a  —  bi=  rfcos  (—  <£)  +  /sin  (—  <£)]  =  r(cos  <f>  —  /sin  <£) 

is  called  the  complex  number  conjugate  to  a  +  bi.  Its  geometrical 
representation  is  (cf.  Fig.  2)  the  reflection  of  the  point  a  +  bi  on 
the  real  axis.  The  number  conjugate  to  the  conjugate  is  the 
original  number.  The  conjugate  of  the  opposite  (IV,  §  2)  is 
the  opposite  of  the  conjugate. 

*  Amplitude  is  used  by  the  translator  instead  of  arcus  and  is  denoted  when  con 
venient  by  am. —  S.  E.  R. 


i6 


I.    COMPLEX   NUMBERS 


a+zb 


FIG.  2 


§  6.  Geometrical  Representation  of  Addition  and  Subtraction 
of  Complex  Numbers 

From  the  points  which  represent  geometrically  the  two  com 
plex  numbers  a  -+-  bi  and  c  +  di,  we  construct  as  follows  the  point 
which  represents  their  sum  : 

I.  Connect  the  two  given  points  with  the  origin  by  straight  lines 
and  complete  the  parallelogram  thus  determined ;  the  fourth  vertex 
is  the  point  required, 

The  proof  is  obtained  from  Fig.  3,  in  which  the  necessary 
auxiliary  lines  are  drawn.  That  it  also  holds  when  the  points 
do  not  both  lie  in  the  first  quadrant  follows  from  the  agreements 
made  in  §  4  about  the  signs. 

Another  form  of  the  rule  is  the  following : 

II.  Draw  a  line  segment  from  the  point  a  +  bi  in  the  same  di 
rection,  of  the  same  length,  and  parallel  to  the  one  from  o  to  c  +  di  ; 
its  end  point  is  then  (a  +  bi)  +  (c  +  di). 

The  commutative  law  for  addition,  the  properties  of  which 
were  discussed  in  §  2,  follows  from  the  first  form  of  the  above 
rule  and  the  associative  law  from  the  second  form.  Moreover 


§5-    REPRESENTATION   OF   ADDITION   AND   SUBTRACTION      I/ 


we  obtain  from  this  the  first  example  of  using  a  geometrical 
theorem  for  the  purposes  of  analysis.  We  refer  to  the  elementary 
geometrical  theorem  that  in  any  triangle  any  side  is  not  greater  * 


FIG.  3 

than  the  sum  (and  not  less  than  the  difference)  of  the  other  two 
sides.  The  following  important  theorem  having  particular  appli 
cation  in  convergence  proofs  is  obtained  from  this  on  the  basis 
of  definition  III,  §  4  : 

III.  The  absolute  value  of  the  sum  of  two  complex  numbers  is 
not  greater  than  the  sum  (and  not  less  than  the  difference]  of  their 
absolute  value  s.\ 

*  This  form  of  expressing  the  theorem  brings  to  mind  a  limiting  case  which 
must  not  be  excluded  here,  viz.  where  the  triangle  degenerates  to  a  straight  line, 
t  An  algebraic  proof  of  this  theorem  is  the  following  : 


Since 
then 


|  a  +  bi  |  = 


+  £-,     \c-\-di\- 
\a  +  bi  +  c  +  <ii  |2  =  (a  +  O2  + 

(  |  a  +  bi  |  +  |  c  +  di  \  )2  -  |  a  4-  bi  +  c  +  di 


=  2.  V  (a2  +  £-)  (c-  -\-  d-)  —  2  ac  —  2  bd 


ti~d*  -f  b  V2  - 


2  +  2  abed  + 


1 8  I.    COMPLEX   NUMBERS 

Repeated  application  of  the  first  half  of  this  theorem  gives 
the  following  more  general  one : 

IV.  The  absolute  value  of  the  sum  of  an  arbitrary  number  of 
complex  numbers  is  not  greater  than  the  sum  of  their  absolute  values, 

The  geometrical  representation  of  subtraction  is  obtained  by 
reversing  the  construction  given  in  Fig.  3  : 

V.  To   construct  geometrically  the  point  which    represents    the 
difference    (a  +  bi~)  —  (c  +  *#),    connect   the  points    (a  +  bi}    and 
—  (c  +  di)  with  the  origin  by  straight  lines  and  complete  the  paral 
lelogram  thus  determined ;  its  fourth  vertex  is  the  point  required. 

Or: 

Let  a  +  bi  be  the  origin  of  a  line  segment  which  is  parallel  and 
equal  in  length  but  oppositely  directed  to  the  one  drawn  from  o  to 
c -\-  di  ;  its  end  point  is  then  the  required  point. 

§  6.   Geometrical  Representation  of  Multiplication  of  Complex 
Numbers 

The  definition  of  the  product  of  two  complex  numbers  given 
in  §3,  equation  (n)  may,  according  to  the  results  of  §4,  be 
put  in  a  form  by  means  of  which  the  product  can  be  con 
structed  geometrically.  Let 

a  -f-  bi  =  r±  (cos  ^  -f-  /  sin  ^J, 
c  +  di  =  r2  (cos  <£2  +  i  sin  <£2), 
But 

(«</_&)  2  2>Q, 

and  accordingly 

&&  +  W2.*abcd\ 

therefore  the  expression  in  [  ]  is  not  negative  since  the  first  root  is  to  be  taken  as 
positive  (the  second  root  might  be  taken  positive  or  negative). 
Hence 

|«  +  bi  |  +  |  c  +di  |  ^  |  #+  bi  +  c  +  di\.  Q.E.D. 


§  6.    REPRESENTATION   OF   MULTIPLICATION  1  9 

and  therefore 

(a  +  bt]  (c  +  <#)  =  r!  •  r2[(cos  <fo  cos  <f>2  -  sin  <fo  sin  <£2) 
+  /  (sin  <fo  cos  <£2  +  cos  <fo  sin 


But  by  the  addition  theorem  for  trigonometric  functions  (A.  A. 
§^74)  this  is 

(1)  =  /ir2  [cos  (<fo  +  <fe)  +  *  sin  (<fo  +  <fo)]. 

The  following  theorem  is  thus  evident  : 

I.  The  absolute  value  of  a  product  is  equal  to  the  product  of  the 
absolute  values  of  the  factors,  tJie  amplitude  is  equal  to  the  sum  of 
the  amplitudes  of  the  factors. 

We  have  seen  in  §  3  that  the  product  of  two  complex  num 
bers  is  single-valued  ;  then  the  value  of  this  product  must  be 
the  same  whichever  ones  of  the  infinitely  many  values  of  the 
amplitude  of  the  separate  factors  (VII,  §  4)  we  select.  In  fact 
this  is  evident  directly  :  if  we  increase  the  amplitude  of  a  factor 
by  an  arbitrary  multiple  of  2  TT,  the  amplitude  of  the  product  in 
creases  by  the  same  multiple  of  2  TT  and  hence  the  value  of  the 
product  itself  is  not  changed. 

A  special  case  worthy  of  notice  is  that  for  which  r2  =  r±  and 
<fo=  ~<£i>  viz., 

(2)  (a  +  bi)(a-bi)=<*  +  P\ 

in  other  words  : 

II.  The  product  of  two  conjugate  complex  numbers  is  real  and 
equal  to  their  common  norm. 

That  the  associative  and  the  commutative  laws  hold  for  a 
product  is  at  once  evident  from  equation  (i).  Furthermore, 
this  equation  enables  us  to  construct  a  product  geometrically. 
In  Fig.  4,  let  c  be  the  point  which  represents  the  product  ab 


20 


I.    COMPLEX   NUMBERS 


O 


geometrically  ;  and  if 


then,    in    accordance    with 
equation  (i), 


FIG.  4 


and  thus 

^  boc  =  <J>i  =  ~%.ioa  ; 

and          01  :  oa  =  ob  :  oc. 


Hence  the  triangles  oia  and  obc  are  similar  to  each  other 
in  all  respects  and  the  required  construction  is  the  following : 

III.  If  a,  b  are  the  points  which  represent  geometrically  the 
numbers  to  be  multiplied,  construct  the  triangle  obc  similar  in 
all  respects  to  the  triangle  oia  ;  the  third  vertex  c  of  this  triangle 
represents  the  product  ab. 

In  this  construction  we  use  in  addition  to  the  origin  the  unit 
point  on  the  #-axis  ;  this  was  not  the  case  for  the  construc 
tion  of  a  sum  in  Fig.  3. 

§  7.    Division  of  Complex  Numbers 

I.  The  quotient  of  two  complex  numbers  a  :  b  is  defined  as  that 
complex  number  c  which,  when  multiplied  by  b,  gives  a. 

Following  the  method  of  representing  a  product  as  given  in 
I,  §  6,  the  solution  of  the  problem  indicated  by  this  definition  is 


§  7.    DIVISION  OF  COMPLEX   NUMBERS  2 1 

at  once  evident ;  thus 

(!)  2  =  3  [cos  (^  -  <£2)  +  i  sin  (<fc  -  «], 

b      r2 
and  hence 

II.  The  absolute  value  of  the  quotient  of  two  complex  numbers  is 
equal  to  the  quotient  of  their  absolute  values  ;  its  amplitude  is  equal 
to  the  difference  of  their  amplitudes  (equal  to  the  angle  boa\ 

Here  also  the  many-valuedness  of  the  amplitude  has  no  effect 
upon  the  result.  For,  if  we  increase  the  value  first  selected  for 
the  amplitude  of  the  dividend  or  of  the  divisor  by  2  TT,  the 
amplitude  of  the  quotient  increases  or  decreases,  respectively, 
by  2  TT  according  to  the  above  rule  for  determining  the  amplitude 
of  a  quotient.  Neither  the  one  nor  the  other  changes  the  value 
of  the  result.  Therefore, 

III.  The  division  of  two  complex  numbers  is  always  a  possible, 
single-valued,  definite  operation  excepting  the  case  where  the  divisor 
is  equal  to  zero. 

Returning  now  from  this  trigonometrical  representation  of 
complex  numbers  to  the  original,  we  find 

,   .  a  +  #  =  (ay  +  £8)  +  (-  «8  +  fly>' 

y  +  8*  y2  +  S2 

On  investigation  of  the  special  case  where  ?\  =  r«,  <j>-2=  —  <£i, 
we  find  that 

IV.  The  quotient  of  two  conjugate  complex  numbers  is  a  number 
whose  absolute  value  is  I . 

V.  The  quotient  of  I  by  a  complex  number  a  is  called  (as  with 
real  numbers)  the  reciprocal  of  a. 

If  a  =  a  -f  fti  =  r  (cos  $  -f  /  sin  <£) 

then 

(3)  --3^»£(cwr#-/ain». 

a      «-  -f-     ~      r 


22  I.    COMPLEX  NUMBERS 

From  these  formulas  it  is  evident  that,  as  with  real  numbers, 
the  following  theorem  holds  : 

VI.  A  complex  number  is  divided  by  another  when  it  is  multi 
plied  by  the  reciprocal  of  the  latter. 

It  therefore  follows  for  division  of  complex  numbers,  that 
all  the  rules  for  manipulating  are  valid  just  as  with  real  num 
bers.  This  result  and  the  results  of  the  preceding  paragraphs 
are  stated  in  the  following  theorem : 

VII.  In  the  field  of  the  four  fundamental  operations  we  may 
operate  with  complex  numbers  as  with  real  numbers. 

It  is  important  that  we  make  the  meaning  of  this  state 
ment  entirely  clear.  It  contains  at  once  the  proposition  that 
there  are  combinations  of  number -pairs,  which  obey  the 
same  rules  as  the  combinations  of  single,  real  numbers 
designated  by  the  names  addition,  subtraction,  etc.  It 
contains  further  the  conventional  agreement  that  we  shall 
retain  for  these  combinations  the  same  name  and  the  same 
symbol  which  are  already  used  for  such  combinations  of  single 
numbers. 

There  are  no  corresponding  theorems  for  triple  numbers, 
quadruple  numbers,  etc.  It  is  possible  to  give  combina 
tions  of  these  —  and  indeed  in  many  ways  —  which  obey 
nearly  all  the  laws  for  operating  with  real  numbers ; 
but  there  are  no  such  numbers  which  can  be  combined 
according  to  all  of  these  laws.  The  proof  of  this  theorem 
is  not  within  the  scope  of  this  book.*  We  cannot  even 
discuss  the  question  whether  or  not  it  is  desirable  from 
certain  points  of  view  to  introduce  such  "  higher  complex 

*  Cf.  D.  HILBERT,  Gott.  Nach.,  1896,  or  O.  STOLZ  and  A.  GMEINER,  Theo- 
retische  Arithmetik,  Lpz.  1902,  Chap.  X. 


§  7.    DIVISION   OF  COMPLEX   NUMBERS  23 

numbers"*  and  whether  they  would  at  once  obey  the  laws  of 
general  arithmetic.  t 

If  the  order  in  elementary  algebra  is  to  be  followed  in  the 
development  of  these  numbers,  we  should  take  up  next  the  so- 
called  operations  of  the  third  grade,  raising  to  powers,  extrac 
tion  of  roots,  and  the  finding  of  logarithms  ;  but  we  shall  postpone 
the  discussion  of  them  to  later  chapters  (§§  18,  56,  63)  and  at 
present  seek  new  results  in  the  field  of  the  four  fundamental 
operations  by  applying  to  the  conceptions  of  algebra  the  notions 
of  a  variable  quantity  and  of  function  belonging  to  analysis 
(A.  A.  §  19). 

EXAMPLES 

1.  Show  that  (cos  6  +  i  sin  0)n  =  cos  nO  +  /sin  n9  for  n  posi 
tive  or  negative,  integral  or  fractional. 

2.  Put  the  following  expressions  in  the  form  r  (cos  0  +  /sin  0)  : 


(cos  * 
A  -f  sin04-gcos0y 
\i  +  sin0  —  icosOJ  ' 

3.  Find  all  the  values  of 

(a)  i*;        ft  (-/)*;  to  (VI-O1; 

(J)  /*;          00  (i+/V3)*;        (/)  32*- 
Find  the  values  of  V/  in  the  form  x  +  iy,  x  and  y  real,  and 
represent  them  graphically. 

4.  Find  and  represent  graphically  the  cube   roots  of  unity. 
Show  that  their  sum  is  zero  and  that  they  form  a  geometrical  series. 

Show  also  that  the  n  nth  roots  of  unity  form  a  geometrical  series. 

*  An  article  by  CHAPMAN,  Bulletin  N.Y.  Math.  Society,  Vol.  I,  p.  150,  entitled 
"  Weierstrass  and  Dedekind  on  higher  complex  Numbers"  may  be  of  interest  to 
the  reader  from  the  point  of  view  of  the  general  theory.  —  S.  E.  R. 

t  Cf.  H.  HANKEL,  Theorie  der  complexen  Zahlensysteme,  Lpz.  1867;  also 
S.  LIE,  Kontinuierliche  Gruppen,  edited  by  G.  Scheffers,  Lpz.  1893,  chap.  21. 


24  I.    COMPLEX   NUiMBERS 

5.    Show  that  the  two  lines  joining  the  points  z  =  a,  z  =  b 
and  z  =  c,  z  =  d  will  be  perpendicular  if 


that  is,  if  f  -    —  j  is  purely  imaginary.     Find  the  condition  that 
these  two  lines  shall  be  parallel.     (Cf.  I,  §  9  and  following.) 

6.    Show  that  if   M  is    the    middle   point  of   CD,    OM  = 
OD)  in  which  O  is  the  origin  in  the  complex  plane. 


7.  Let  A  and  B  be  any  two  points  in  the  complex  plane  ;  we 
wish  to  find  the  complex  quantity  represented  by  AB.    Connect 
each  point  with  the  origin  O.     Then,  according  to  the  definition 
and  construction  of  the  difference  of   two  complex  quantities, 
AB  is  equal  to  OB  —  OA  (where  AB  means  from  A  to  B}. 

8.  Given  four  points  A,  B,  C,  D  on  the  axes  at  unit  distance 
from  the  origin.     Find  the  complex  quantities  which  represent 
the  distances  AB,  BC,  CD,  DA. 

9.  What  complex  quantities   represent  the   vertices  of  the 
hexagon  inscribed  in  the  unit  circle  about  the  origin  ?     What 
complex  quantities  represent  the  sides  of  this  hexagon  ? 

10.  Given  any  three  points  A,  B,  C  in  the  plane  such  that 
OA,  OB,  OC  are  non-collinear.  To  prove  that  it  is  always 
possible  to  express  the  relation  between  them  in  the  form 

OC  =  a  -  OA  +  b  -  OB  (a  and  b  are  real)  ; 
and  further  that  this  representation  is  itnique. 

PROOF.  —  Draw  MC  parallel  to  OB.  In  the  triangle  OMC,  OM  +  MC  = 
OCby  addition.  But  OM  '  —  a  •  OA  since  OM  and  OA  differ  only  in  abso 
lute  value  ;  likewise  MC  =  b  •  OB.  It  follows  that 

OC=a>  OA  +  b.  OB. 


§  7.    DIVISION   OF   COMPLEX  NUMBERS  2$ 

Next,  suppose  a  second  rep 
resentation 


OC  =  a'  -  OA  +  V  -  OB. 
Therefore 
(a -a'} 


but  this  means  that  O,  A.  B  are          ^-^"^^        ^ 

0^^ 

collinear,    contrary    to    the    hy 
pothesis,  since  then  OA  is  equal  to  a  constant  times  OB.     Hence  the  repre 
sentation  is  unique. 

11.  Construct  w  =  1/0,  i.e.  iv  :  i  =  i  :  z. 

12.  Multiply  2  times  3  graphically.     Illustrate  WEIERSTRASS' 
definition  of  multiplication  that  to  multiply  a  by  b,  operate  on  b 
as  was  done  on  unity  to  get  a. 

a  4-  ib 


13.    Prove  geometrically  that 


a 


=  i, 


14.  Find  the  points  which  divide  the  line  segment  from  o  to 
i  in  the  ratios  ±  (i  -f- 1). 

15.  Show  by  expanding  (cos  B  +  /sin  0)n  and  equating  reals 
and  imaginaries  that 

cos  nO  =  cos" 0  -  n'  n  ~  *  cosn~2 0 sin2 0  -\ 

2 

sin  nO  =  n  cos71"1  0  sin  0  —  n  '  n  ~  I  '  n  ~  2  cosn~3  0  sin3  6+  ••• 

i  -2-3 

16.  Show  that 

(a)  sin  3  0  =  3  sin  0  cos2  0  —  sin3  0  =  3  sin  0  —  4  sin3  0 ; 

(b)  cos  30=  cos3  0  —  3  cos  0  sin2  0  =  4  cos3  0—3  cos  0  ; 

(c)  cos  4  0  =  cos4  0  —  6  cos2  0  sin2  0  +  sin4  0. 

Find  similar  expressions  for  sin  40,  cos  5  0,  cos  6  0,  sin  7  0. 

17.  If    cos  0  +  /  sin  0  =  #,     show    that    2  cos  ;*0  =  xn  +  — , 


26  I.  COMPLEX  NUMBERS 

2  i  sin  nd  =  xn  --  .     Show  also  that 

xn 

(2  cos  ey  =  (W  £)"=  (*>  +  -L 

=  2  cos  n&  +  2  n  cos  («  —  2)  0  H  ----  . 

Derive  a  similar  expression  for  sinn  0,  discussing  the  cases  for 
n  even  and  odd. 

18.    Expand  cos50  in  a  series  of  cosines  of  multiples  of  0 
HINT.  —  Put  (2cos6)$  = 


* 

=  2  COS  50  +  .... 

Expand  similarly  the  following  expressions  : 

sin50,    cos70,    sin8  (9,    cos5  0  •  sin7  0. 

19.  Show  that 

'  (x  +  y<»3  +  so>32)  (x  +  _yw32  +  zo>3)  =  x2+y*  +  z*  —  yz  —  zx  —  xy. 

20.  Prove  that  (.#2n  —  2^ntf"cos0  -f  a2n)  =fx2  —  2xacos- 


« 

HINT.  —  Make  use  of  the  formula 

(yzn  _  2  xnan  cos  0  +  «2n)  =  {^n  —  #n  (cos^+zsin^)}{^n—  an  (cos  0—  t'sin  (?)}, 
resolving  each  of  the  last  two  expressions  into  n  factors. 

21.  Show  that  the  triangle  xyz  is  equilateral  if 

x2  +  y2  -{-  z2  —  xy  —  yz  —  zx  =  o. 
HINT.  —  If  ABC  be    the  triangle,   the  line  segment    CA  is  BC  turned 

through    an     angle  —  ;     and,    since    cos  --  M'  sin  —  =  w3    and    cos  — 

3  3  o  J 

-  zsin—  =  —  =  w32,    we     have     *-z=(>—  /)  o>3     or     (x—  z)  =(z—  y)  o>32. 

3        ws 
Hence  ^  +  _yw3  +  2o>32  =  o,  etc.,  and  the  result  follows  from  Ex.  19. 

22.  Show   that    \a  -f  ^|2  -f  \a  -  &\2  =  2\a  j|2  +   £  2J.     How   is 
this  related  to  the  geometrical  theorem  that,  if  M  is  the  middle 
point  of  PQ  and  O  is  the  origin,  ~OP2  +  ^2  =  2  <9J/2  +  2 


§  7.    DIVISION  OF  COMPLEX  NUMBERS  27 

23.  Show  that  the  roots  of  8  x3  —  4  x-  —  4  x  +  i  =  o  are 

cos-,    cos  3^,    cos  5-E. 

7  7  7 

HINT.  —  Put  (cos  0  +  z  sin  #)"  =  —  I,  z'.^.  cos  7  0  =  —  I.     From  this 

0  =  1   2z   ir.  zr,  i^?  iu:?  uzr. 

77        7        7        7         7    '      7 

Expand  (cos0  -f  *sin0)7  =  —  I  by  the  binomial  theorem  and  equate  the 
real  parts  on  each  side.     This  gives 

COS7  e  —  21  cos5  6  sin5  e  +  35  cos3  6  sin4  6  —  7  cos  0  sin6  d  +  i  =  o,    i.e. 
(i)  64  cos7  6  —  1 12  cos5  6  +  56  cos3  0  —  7  costf  +  i  =  o, 

and  its  roots  are 

cos-,    cos  2-^,    cos  5-^,    Cos  2-??    COs2-??    cos — — ,    cos    ^  ^ . 

777  7  7  7  7 

since  (i)  is  the  original  equation  cos  j  6  =—  i.     But 

7  7T  Q  IT  ?  7T  I  I  7T  7  IT  I  7  7T  7T 

cos  « —  =  —  i ,    cos  - —  =  cos  >2 —  ,    cos =  cos  — — ,    cos    ^     =  cos  —  . 

7  777777 

The  roots  of  (i)  are  thus  :  —  I,  and  cos  — ,  cos  ^,  cos  5£  repeated  twice. 

Now  put  x  =  cos  6  and  (i)  becomes 

(x  +  i)  (8-r3  -  4*2  -  4 *  +  1)2  =  o. 

Thus  cos  - ,  cos  —  ,  cos  5_-  are  the  roots  of 
7  7  7 

8  x3  —  4  .r2  —  4  *  +  i  =:  o. 

24.  Show  that  the  two  sets  of  points  a,  b,  c  and  x,  y,  z  in  the 
complex  plane  form  similar  triangles  if 


a,  x,     i 

b,  y,      i 


=  o. 


25.    If  the  points  x,  y,  z  are  collinear,  show  that  real  numbers 
/,  q,  r  can  be  found  such  that 

/  +  g  -f-  r  —  o  and  /AT  -f  qy  +  rz  =  o. 

HINT.  —  Use  the  condition  for  the  similarity  of  the  triangle  xyz  and  a 
certain  other  triangle  on  the  real  axis. 


CHAPTER   II 

RATIONAL  FUNCTIONS  OF  A  COMPLEX  VARIABLE  AND  THE 
CONFORMAL  REPRESENTATIONS  DETERMINED  BY  THEM 

§  8.   General  Introduction ;  the  Function  z  +  a  and  the  Parallel 
Translation 

IN  this  and  the  following  paragraphs  we  discuss  the  case 
where  one  of  the  two  complex  mumbers  to  be  combined  by  our 
elementary  operations  is  regarded  as  fixed  (constant),  the  other 
as  variable,  that  is,  as  a  symbol  which  is  supposed  to  take  differ 
ent  values  throughout  the  course  of  the  investigation.  We  state 
more  definitely  that  we  consider  this  variable  as  a  number  of 
unlimited  variation,  that  is,  as  a  symbol  which  may  be  regarded 
as  representing  any  complex  quantity  whatever.*  We  exhibit 
this  distinction  in  the  notation  in  that  we  follow  the  usual  custom 
of  designating  constant  quantities  by  the  first  and  variables  by 
the  last  letters  of  the  alphabet. 

This  difference  between  constants  and  variables  is  clearly 
displayed  in  the  form  of  an  equation  between  two  different 
variables.  To  begin  with  the  simplest  example,  let  us  put 

(i)  z'  =  z  +  a-} 

thus  z  and  z'  are  both  quantities  which  vary  together.  The 
variation  of  z',  however,  depends  in  a  definite  way  upon  the 
variation  of  z  according  to  the  equation  (i) ;  on  this  account  it 

*  Thus  a  variable  is  a  symbol  which  represents  any  one  of  a  set  of  numbers 
while  a  constant  is  a  special  case  of  a  variable  where  the  set  consists  of  but  one 
number.  For  this  and  the  definition  and  history  of  a  function,  cf.  VEBLEN  and 
LENNES,  Infinitesimal  Analysis  (Wiley  and  Sons) ,  p.  44.  —  S.  E.  R. 

28 


§  8.   THE   FUNCTION   z  +  a  2Q 

is  called  a  function  *  of  z  and  in  fact  a  rational  function.  We 
define  as  follows  : 

•  I.  A  complex  variable  z1  is  called  a  RATIONAL  FUNCTION  of 
another  variable,  z,  when  it  is  possible  to  deduce  it  from  z  and  con 
stants  by  a  finite  number  of  additions,  subtractions,  multiplications, 
and  divisions. 

II.  If  there  are  no  quotients^  in  this  function  it  is  called  a 
rational  INTE  ORAL  function. 

If  we  represent  z  and  z'  geometrically  in  two  different  planes, 
then  an  equation  of  the  form  : 

(2)  ,'=/(*) 

determines  a  representation  \  (a  map)  of  the  s-plane  on  the 
s'-plane  in  such  a  manner  that  to  each  point  z  of  the  first  plane, 
there  corresponds  a  point  z'  of  the  second.  If,  however,  we 
represent  z  and  z'  on  the  same  plane  and  refer  them  to  the  same 
axes,  then  by  such  an  equation  each  point  z  of  the  plane  is  set 
in  correspondence  with  a  definite  point  z'  of  the  same  plane.  It 
represents,  as  we  say,  a  transformation  of  the  plane  into  itself. 

In  regard  to  equation  (i)  in  particular,  it  is  shown  in  Fig.  3 
that  each  point  z'  is  obtained  from  the  corresponding  point  z 
by  moving  the  whole  plane  parallel  to  itself  in  the  direction 
and  the  length  of  the  line  segment  oa : 

*  We  shall  later  (in  $  33)  define  the  phrase  "  Function  of  a  Complex  Variable  " 
in  a  sense  more  restricted  than  the  one  given  here.  It  will  then  be  shown  that  the 
rational  functions  to  be  treated  in  this  chapter  are  also  in  that  restricted  sense 
"  Functions  "  of  their  arguments.  For  this  reason  it  is  allowable  to  define  "  Rational 
Function  of  z  "  without  giving  a  previous  formal  definition  of  what  is  in  general 
understood  by  "  Function  of  z." 

t  Only  such  divisions  are  to  be  excluded  for  which  the  denominator  depends 
upon  z.  Division  by  a  constant  is  multiplication  by  its  reciprocal,  that  is,  by  a  con 
stant  (A.  A.  §  20). 

J  This  idea  of  the  geometrical  representation  of  the  dependence  of  z'  on  z  as  a 
transformation  or  a  mapping  of  the  z'-plane  on  the  z-plane  is  due  to  RlEMANN, 
Grundlagen  fur  eine  allgemeine  Theorie  der  Funktionen,  Werke,  p.  5.  —  S.  E.  R. 


30  II.    RATIONAL   FUNCTIONS 

III.  The  transformation  of  the  plane  into  itself  determined  by 
equation  (i)  is  a  parallel  displacement  (a  translation)  in  direction 
and  distance  equal  to  oa. 

In  particular, 

IV.  When  the  relation  between  z  and  z'  is  represented  by  equa 
tion  (7),  any  figure  formed  by  d -points  is  congruent  to  the   one 
formed  by  the  corresponding  z-points. 

§  9.  The  Function  az 

The  next  simple  rational  function  of  z  to  be  considered  is  the 
product  of  z  and  a  constant  a,  viz. : 

(1)  z1  =  az. 

We  ask  now  what  kind  of  transformations  of  the  plane  into  itself, 
that  is,  what  kind  of  mappings  of  the  s-plane  on  the  z'-plane, 
are  possible  by  means  of  this  equation  ?  We  exclude  first  of 

all  the  case 

0  =  0, 

since  in  this  case  z'  =  o  whatever  the  value  of  z  may  be,  and 
consequently  there  is  no  proper  transformation.  We  now  con 
sider  two  important  special  cases  : 

First:  let  a  be  a  number  whose  absolute  value  is  i  ;  then 

(V,  §  4) 

(2)  a  =  cos  a  +  i  sin  a, 

in  which  a  is  a  real  angle.  The  graphical  construction  j>f  a 
product  given  in  Fig.  4,  §  6,  shows  that  the  line  segment  oz'  is 
equal  in  length  to  the  line  segment  ~oz,  but  that  it  makes  with 
the  *-axis  an  angle  increased  by  the  constant  a.  Each  point  z' 
will  then  be  obtained  from  its  corresponding  point  z  by  rotating 
the  whole  plane  about  the  origin  through  the  angle  a ;  in  other 
words : 


§9-    THE   FUNCTION  az  31 

I.  The  transformation  of  the  plane  into  itself  by  means  of  the 
equation  g,  =  (CQS  ft  +  .  sjn  ^  (a  m?/) 

w  effected  by  rotating  it  about  the  origin  through  the  angle  a. 

(In  particular,  s'  =  iz  determines  a  rotation  through  a  right 
angle,  z'  =  —  z  a  rotation  through  two  right  angles.) 

Moreover, 

II.  /;/  this  case  (as  in  §  8),  the  figures  formed  from  the  z' -points 
are  congruent  to  tJie  ones  formed  by  the  corresponding  z- points. 

Second:  let  a  be  a  real  positive  number  r.  Then  any  point  z 
and  its  corresponding  point  z'  lie  on  a  straight  line  through  the 
origin  (cf.  Fig.  4)  and  are  so  related  that  the  length  of  the  line 
segment  oz1  has  a  constant  ratio  to  that  of  oz.  We  say : 

III.  The  transformation  of  the  plane  into  itself  by  means  of  tJie 

equation  , 

y  z'  =  rz 

(in  which  r  is  real  and  positive]  is  effected  by  stretching  it  from  the 
origin. 

(If  r  <  i ,  the  distance  from  the  origin  is  shortened  instead  of 
lengthened ;  the  word  "  stretching"  will  include  both  cases.) 

IV.  Any  figure  formed  by  the  z' -points  bv  means  of  this  trans 
formation  is  not  congruent  to  the  one  formed  by  the  corresponding 
z-points,  but  is  similar  to  it. 

Having  thus  disposed  of  these  two  special  cases,  let  us  con 
sider  the  general  case  of  the  transformation  (i).  Multiplication 

a  =  r  (cos  a  +  /"  sin  a), 

obeys  the  associative  law  for  products  (equation  (12),  §  3) ;  we . 
may  therefore  first  multiply  by  r  and  then  by  (cos  «  +  *  sin  a). 
Consequently, 


32  II.    RATIONAL   FUNCTIONS 

V.  The  general  transformation    (/)    is   effected  by  a    rotation 
about  the  origin  through  the  angle  a  (=  the  amplitude  of  a)  and  a 
stretching  from  this  point  in  the  ratio  \a    :  I.     It  is  a"  similarity  " 
transformation  with  the  origin  as  center  of  the  similarity  (that  is, 
figures  are  transformed  into  similar  figures*}* 

Moreover,  from  the  commutative  law  for  multiplication  we 
have  the  theorem  that  the  result  is  independent  of  the  order  of 
the  operations  of  stretching  and  rotating.  It  is  customary  to 
express  it  in  this  way : 

VI.  Stretching  from  a  point  and  rotating  about  it  are  permut- 
able  operations. 

§  10.  The  Linear  Integral  Function  and  the  General  "  Similarity  "  f 
Transformation 

I.  If  a  complex  variable  z'  depends  upon  another,  z,  according 
to  the  relation 

(1)  z'  =  az  +  b, 

in  which  a,  b  are  arbitrary  complex  constants,  then  we  say  that  z' 
is  a  linear  integral  function  of  z. 

As  in  §  9  we  exclude  the  case  a  =  o ;  for,  in  this  case  equa 
tion  (i)  represents  no  proper  transformation  of  the  plane  into 
itself,  since  the  fixed  point  5'  =  b  corresponds  to  an  arbitrary 
point  z  (that  is,  the  2-plane  is  transformed  into  the  fixed  point 
*'  =  £).*  But  if 

(2)  a  =£0, 

transformation  (i)  can  be  compounded  in  various  ways   from 
the   simpler   transformations    already  discussed.     We    can,  for 

*  Words  in  the  parenthesis  inserted  by  the  translator, 
t  Cf.  V,  §  9,  and  V,  §  10.  — S.  E.  R. 


§  io.    THE   LINEAR   INTEGRAL   FUNCTION  33 

example,  introduce  the  auxiliary  variable  2"  by  the  equation 

(3)  *"  =  <>*•> 

it  then  follows  that 

(4)  z'  =  z"  +  b. 

II.  The  transformation  determined  by  (i)  is  therefore  obtained 
by  stretching  from  the  origin  and  rotating  about  it  (equation  j),  fol 
lowing  with  a  translation  (equation  4). 

But  we  might  also  introduce  first  an  auxiliary  variable  z'" 
(always  assuming  equation  2)  by  the  equation 

(5)  Z'"  =  Z  +  a< 
in  consequence  of  this 

(6)  ,•  =  „»'. 

III.  Therefore,  the  general  transformation  (i)  is  also  performed 
by  first  displacing  the  plant  parallel  to  itself,  then  stretching  and 
rotating. 

In  this  connection  we  notice  that  the  coefficient  of  stretching 
and  rotating  in  (6)  is  the  same  as  in  (3),  but  that  the  coefficients 
of  the  parallel  translations  in  (4)  and  (5)  agree  only  for  0  =  1, 
that  is,  when  there  is  no  stretching  and  rotating.  We  now  state 
this  explicitly  as  follows  (cf.  Theorem  VI,  §  9) : 

IV.  Parallel  translation  on  the  one  hand,  stretching  and  rotat 
ing  on  the  other,  are  not  permutable  operations. 

We  come  now  to  a  third  important  representation  of  the 
transformation  of  the  plane  into  itself  by  means  of  (i)  by  dis 
cussing  the  question  whether  there  are  definite  points  which 
remain  fixed  for  this  transformation,  that  is,  expressed  analyti 
cally,  whether  there  are  values  z  which  coincide  with  the  values 
2'  corresponding  to  them  according  to  (i).  For  every  such  value, 

(7)  z  =  az  +  b; 


34  II.    RATIONAL   FUNCTIONS 

for  a  •=£  i ,  this  equation  has  one  and  only  one  root ;  denote  it 
by  £  and  we  obtain : 

<">  <-£;• 

By  substituting  the  value  of  b  from  equation  (8)  in  equation 
(i),  it  becomes : 

Transformation  (i)  can  therefore  be  performed  by  applying 
successively  the  three  simpler  transformations  : 


in  other*  words,  we  so  translate  the  plane  parallel  to  itself  that 
the  fixed  point  z  =  £  coincides  with  the  origin  ;  we  then  apply 
a  stretching  from  this  point  and  a  rotation  about  it  ;  and 
finally  return  this  point  to  its  first  position  z  =  £.  But,  as  is 
evident  geometrically,  we  obtain  the  same  result  if  we  stretch 
from  the  point  z  =  £  and  rotate  about  it.  Hence  the  following 
theorem  : 

V.  If  a  =£  i  the  transformation  of  the  plane  into  itself  accord 
ing  to  (/)  is  performed  by  rotating  through  the  amplitude  of  a 
about  the  point  : 


and  stretching  from  this  point  in  the  ratio  \a\\i.  In  this  way 
each  figure  is  transformed  into  a  similar  one. 

We  can  also  obtain  the  same  result  in  a  somewhat  different 
manner.     An  equation  of  the  form 

(10)  Z=/(z) 

can  be  given  a  different  interpretation  from  that  in  §  8.  In 
stead  of  regarding  Z  and  z  as  complex  numbers  which  belong 
to  two  different  points  of  the  plane  in  reference  to  the  same  sys- 


§  io.    THE   LINEAR   INTEGRAL   FUNCTION  35 

tern  of  coordinates,  we  can  look  upon  this  equation  as  assigning 
anotJier  complex  number  Z  to  that  point  which,  for  a  definite  sys 
tem  of  coordinates,  represents  the  complex  number  z.  For  a 
particular  form 

(n)  Z=z-£ 

of  equation  (io),  this  number  Z  can  be  defined  as  follows:  By 
making  the  point  called  £  in  the  old  system  the  origin  of  a  new 
system  of  coordinates,  whose  axes  are  parallel  to  the  old  and 
whose  unit  of  length  is  the  same  as  in  the  old  system,  we  obtain 
the  point  which  is  called  z  in  the  old  [Z  in  the  new]  system  of 
coordinates.  The  point  called  z1  in  the  old  coordinates  is,  in 
the  new  system,  assigned  to  the  number: 

(12)  Z' =  *'-£. 

The  relation  between  the  points  z  and  2',  expressed  in  the  old 
system  of  coordinates  by  equations  (i),  (9),  is  expressed  in  the 
new  system  by  the  equation  : 

(13)  Z'  =  *Z, 

that  is,  it  is  a  "  similarity  "  transformation  whose  center  of  simi 
larity  is  called  o  in  the  new  system  and  £  in  the  old.  We  have 
thus  deduced  Theorem  V  in  a  new  way. 

But  we  can  also  prove  the  converse  of  this  theorem.  For,  a 
"  direct  "  similarity  transformation  of  the  plane  is  determined 
whenever  the  points  z±,  z2'  corresponding  to  two  different 
points  z±,  z2  are  given ;  every  third  point  z3  then  has  its  corre 
sponding  point  z3'  fixed  uniquely  from  the  fact  that  the  tri 
angles  z&Zs  and  z^z^z^  must  be  similar  throughout,  including 
the  sense  of  corresponding  angles.  But  we  can  always  deter 
mine  a  transformation  of  the  form  (i)  which  transforms  z^  zz 
respectively  into  %',  z2'.  For  this  purpose  it  is  only  necessary 
that  a  and  b  satisfy  the  equations  : 

(14)  zj=azi  +  b,     zJ=az«  +  b; 


36  II.    RATIONAL   FUNCTIONS 

but  from  these  equations  the  values : 


are  finite  and  determinate  providing  £2  =£  ZA  ;  we  can  say  : 

VI.  Every  transformation  of  the  plane  into  itself,  which  trans 
forms  any  figure  of  the  plane  into  a  similar  one,  including  the  sense 
of  corresponding  angles,  is  expressed  in  the  form  (i).* 

For  the  purposes  of  later  applications  we  express  the  condi 
tions  for  the  similarity  of  two  triangles,  including  the  sense  of 
corresponding  angles,  in  terms  of  the  complex  numbers  repre 
senting  their  vertices.  If  the  triangles  ZyZfa  and  SiVV  are 
similar,  then  the  values  of  a  and  b  in  (15)  must  also  satisfy  the 

equation  : 

zj  =  az,  +  b; 

this  is  true  if  and  only  if 


In  this  equation  we  can  put  the  letters  with  primes  on  one  side 
and  those  without  primes  on  the  other  side  of  the  equality  sign 
and  formulate  the  theorem  as  follows : 

VII.  The  necessary  and  sufficient  condition  for  the  similarity  in 
all  of  their  parts  of  two  triangles  z&z^  and  z^z^z^  is,  that  the 
quotient 


We  can  obtain  the  same  result  geometrically.  For,  the  abso 
lute  value  of  the  difference  z^  —  z±  is,  according  to  V,  §  5,  equal 

*  As  an  example  of  how  geometrical  theorems  result  from  operations  with  com 
plex  numbers,  we  cite  the  theorem  following  from  V  and  VI  : 

Every  direct  similarity  transformation  of  the  plane  which  is  not  merely  a  paral 
lel  translation  can  be  considered  as  a  rotation  about  a  point  which  is  fixed  by  the 
transformation,  and  a  stretching  from  this  point. 


§  io.    THE   LINEAR   INTEGRAL   FUNCTION  37 

to  the  length  of  the  line  segment  from  zl  to  %,  its  amplitude  is 
equal  to  the  angle  which  this  segment  makes  with  the  positive 
half  of  the  real  axes ;  and  similarly  for  the  difference  z3  —  zt. 
The  absolute  value  of  the  quotient  on  the  left  side  of  equation 
(17)  is  then  equal  to  the  ratio  of  the  lengths  of  two  sides  of  the 
triangle  SiZ«z3,  its  amplitude  is  equal  to  the  angle  inclosed  by 
these  lengths ;  and  since  corresponding  interpretations  can  be 
made  for  the  right  side  of  (17),  the  equation  expresses  the  sim 
ilarity  of  the  two  triangles  z^z^  and  ^ 'z2V-  That  the  sense  of 
corresponding  angles  is  also  the  same  follows  from  the  fact  that, 
in  this  kind  of  investigation,  the  amplitude  of  a  complex  number 
has  a  definite  sign. 

Occasionally  equation  (17)  is  used  in  the  determinant  form: 


(18) 


=  o. 


EXAMPLES 

1.  Given  z'  =  iz :    Determine    the    change    effected    by  this 
transformation  in  the  following  figures  (that  is,  determine  the 
figure  in  the  s'-plane  which  corresponds  by  this  transformation 
to  the  following  figures  in  the  s-plane,  the  two  planes  regarded 
either  as  coincident  or  as  separate). 

(a)  The  square  whose  vertices  are  the  points  ±  i  ±  / ; 

(b)  The  unit  circle  whose  center  is  at  the  origin  ; 

(c)  The  triangle  whose  vertices  are  the  points  o,  i  +/,  i  +2  /'. 

2.  Apply  each  of  the  following  transformations  to  the  con 
figurations  of  Ex.  i  and  note  the  change : 

(a)  *'  =  *  +  /; 

(b)  z'  =  z  +  3i; 


38  II.    RATIONAL  FUNCTIONS 

3.  Discuss  the  transformation  z'  =  2  +  (i  +  -\/3  /)  by  putting 

z'  =  x1  -f  ty1  and  z  =  x  +  iy. 

4.  Regarding   the   z'-plane    and    the   z-plane   as  coincident, 
determine  the  configuration  corresponding  to  each  of  the  con 
figurations  of  Ex.  i  for  each  of  the  following  transformations  : 

(a)  z1  =  2  z  ; 

(b)  *'=(l/2)*; 


00    *'=   (l  +  l>+( 

5.  Determine  the  linear  integral  transformations  of  the  form 
z'=  az  +  b  which  transform  : 

00  The  point  —  i  into  the  point  o  and  the  point  o  into  the 
point  +  2  ; 

(/£)  The  point  —  i  into  the  point  o  and  the  point  o  into  the 
point  —  2  ; 

(c]  The  points  i  and  —  *  respectively  into  the  points  +  2  and 
-  2  ; 

6.  Perform  geometrically  the  transformations  in  Ex.  5. 

7.  Perform  the  transformation 


i  st.    By  stretching  and  turning  and  then  rotating. 

2d.    By  translating  and  then  stretching  and  turning. 

3d.  By  reducing  the  equation  to  the  form  z'  —  G  —  a  (z  —  G), 
G  being  the  invariant  point,  transferring  the  origin  to  the  point 
G,  stretching  and  rotating,  etc. 

§  11.    The  Function  -  and  the  Transformation  by  Reciprocal  Radii 

The  investigation  of  the  quotient,  considered  as  a  function  of 
the  dividend,  resolves  itself,  on  account  of  theorem  VI,  §  7, 
into  the  investigation  of  a  product  discussed  in  §  9  ;  considered 


§  ii.    THE   FUNCTION    i/z  39 

as  a  function  of  the  divisor  its  investigation  may  be  referred,  on 
account  of  the  same  theorem,  to  the  case  in  which  the  numera 
tor  is  i.  The  question  then  is:  What  transformation  of  the 
plane  into  itself  is  determined  by  the  function : 

(1)  _  «•=!? 

To  investigate  this,  put 

z  =  x-\-  iy  =  r  (cos  <£  -+-  i  sin  <£), 
z'  =  x'  +  /y  =  r'(cos  <f>'  -f  /  sin  <£')  ; 
according  to  equation  (3),  §  7,  we  therefore  obtain : 

(2)  ^=1,   $'  =  -<!> 
and  therefore          x'  = 


The  transformation  determined  by  these  equations  may  be  re 
garded  as  compounded  from  two  simpler  geometric  transforma 
tions,  each  of  which  considered  by  itself  is  not  determined  by 
rational  functions  of  a  complex  variable.  Let  us  first  introduce 
the  auxiliary  transformation : 

(4)  r=r<    ?  =  -<£, 

or 

(5)  *-*  J=-y- 

And  therefore  to  obtain  transformation  (i),  we  must  after  this  put 

(6)  r1  =  i/V,    <£'  =  <£, 
or 

(7)  *'<        * 

I.  Equations  (4),  (5)  (cf.  VIII,  §  4)  effect  the  transition  from 
any  complex  number  to  its  conjugate,  thus  determining  geometrically 
a  reflection  on  tJie  axis  of  reals. 


40 


II.     RATIONAL   FUNCTIONS 


II.  The  transformation  determined  by  equations  (6)  is  called 
(on  account  of  the  first  one  of  them)  the  transformation  by  recipro 
cal  radii  with  reference  to  the  unit  circle  —  also  called  reflection  * 
on  the  unit  circle.     It  is  important  that  we  investigate  its  most 
essential  properties. 

III.  Transformation  (d)  is  involutoric  ;  that  is,  by  means  of  it 
pairs  of  points  correspond  mutually  to  each  other  ;  (or  more  ex 
plicitly,  if  P  be  transformed  into  P'  and  by  the  same  transfor 
mation    P'  goes    into    P,  the   transformation    is    involutoric).f 
Equations  (6)  are  unchanged  if  r'  is  interchanged  with  ~r  and  <£' 
with  <£.     This  same  property  does  not  follow  so  directly  from 

equations  (7),  but  is 
easily  deduced. 

IV.  If  the  point 
(r,  <£)  lies  outside  of 
the  unit  circle,  the 
point  (r',  <£')  is  the 
intersection  of  the 
chord  of  contact  of 
tangents  from  (r,  <£) 
to  the  unit  circle 
with  the  diameter 
prolonged  through 
(r,  *)  (cf.  Fig.  5). 
The  point  which  corresponds  to  a  point  lying  inside  of  the  unit 
circle  is  obtained  (on  account  of  III)  by  reversing  this  construc 
tion.  Every  point  on  the  unit  circle  corresponds  to  itself. 

A  further  important  property  of   this  transformation  is  that 

*  In  an  applied  sense  the  law  of  reflection  in  optics  is  different.  On  the  other 
hand,  the  transformation  treated  here  is  important  in  electrostatics.  When  used 
there  it  is  spoken  of  as  "  The  Principle  of  the  Thomson  Images." 

f  Words  in  the  parenthesis  added  by  the  translator. 


§  ii.    THE   FUNCTION    i/z  41 

circles  transform  into  circles,  that  is,  all  the  points  on  a  given 
circle  are  transformed  into  points  which  lie  again  on  a  circle. 
For  if  the  coordinates  x,  y  of  a  point  satisfy  the  equation  : 


(8)  tf(*2  +  /)  +  bx  +  cj>  +  d  =  o, 

which  represents  any  arbitrary  circle  for  suitably  chosen  coeffi 
cients,  then,  according  to  (7),  x',  y'  satisfy  the  equation  : 

(9)  <-v'2  +  /2)  +  ^vf  +  cy'  +  a  =  o, 

which  in  general  again  represents  a  circle  except  for  d=  o  when 
it  is  a  straight  line.  The  above  statement  is  therefore  correct 
if  a  straight  line  is  regarded  as  a  special  case  of  a  circle.  The 
following  are  more  precise  statements  of  these  facts  : 

V.  To  a  circle  which  does  not  go  through  the  origin  (a  -=£  o, 
d=£  6]  there  corresponds  a  circle  which  does  not  go  through  the 
origin;  to  a  circle  through  the  origin  (a=f=o,  d=o)  there  corre 
sponds  a  straight  line  which  does  not  go  through  the  origin  ;  a 
straight  line  through  the  origin  (a  =  o,  d=o)  corresponds  to  itself. 

We  also  add  : 

V#.  To  parallel  straight  lines  correspond  circles  with  a  com 
mon  tangent  at  tlie  origin. 

Further,  the  transformation  by  reciprocal  radii  has  the  prop 
erty  that  angles  are  preserved,  that  is,  that  the  angle  of  inter 
section  of  two  curves  is  equal  to  the  angle  of  intersection  of  the 
corresponding  curves.  The  correctness  of  this  statement  can 
be  shown  best  by  considering  first  the  special  case  in  which  one 
of  the  two  curves  is  a  straight  line  through  the  origin.  Thus  if 
PP  and  QQ'  are  two  pairs  of  corresponding  points  (Fig.  6),  it 
then  follows  according  to  equation  (6)  that 

OP-  ~OP'  =  OQ>  OQ'  =  i  ; 
and  hence  A  OPQ  ~  A  OQ'P1  ; 


42  II.    RATIONAL   FUNCTIONS 

and  in  particular : 

(10)  £  OPQ  =  %-OQ'P1. 

If  we  allow  the  point  Q  to  approach  the  point  P  along  a  given 
curve,  then  the  point  Q'  approaches  the  point  P'  upon  the  cor- 


0 


. 

FIG.  6 


responding  curve  ;  PQ,  P'  Q'  become  the  directions  of  the  tan 
gents  to  the  curves,  %  OQ'P'  in  the  limit  will  be  equal  to 
£  O'P'Q',  and  it  therefore  follows  that  in  the  limit 


(n)  £  OPQ  =  %  O'P'Q'.  Q.E.D. 

We  notice  further  in  this  connection  that  the  equality  sign  refers 
only  to  the  absolute  value  of  the  angle  ;  the  two  angles  corre 
sponding  to  each  other  are  opposite  in  sense,  and  the  resulting 
theorem  is  completely  formulated  as  follows  : 

Two  curves  corresponding  to  each  other  form  with  any 
straight  line  corresponding  to  itself,  angles  which  are  equal  but 
of  opposite  sense. 

Moreover,  since  the  angle  between  any  two  lines  is  equal  to 
the  sum  (difference  respectively)  of  the  two  angles  which  the 
two  lines  make  with  a  third,  it  follows  that  : 

VI.  The  angle  in  which  any  two  curves  intersect  is  equal  in  the 
opposite  sense  to  the  angle  of  intersection  of  the  two  curves  which 
correspond  to  the  first  two  by  the  transformation  by  reciprocal  radii. 


§  ii.    THE  FUNCTION   i/z  43 

Since  this  transformation  is  of  frequent  occurrence,  a  name  is 
given  to  it  as  in  the  following  definition : 

VII.  A  transformation,  under  which  the  angle  between  any  two 
curves    is  equal  to  the  angle  betu>een  the  corresponding  curves,  is 
called  a  conformal  representation  *  (also  isogonctl  representation,  or 
a  mapping  with  preservation  of  angles,  or  one  "  similar  in  infini 
tesimal  parts  "). 

Therefore,  according  as  the  sense  of  the  angle  obtained  is 
preserved  or  changed,  we  speak  of  the  representation  as  con- 
formal  "without"  or  "with  inversion  of  angles"  With  this 
terminology  Theorem  VI  is  stated  as  follows : 

VIII.  The  transformation   by    reciprocal  radii  is  a  conformal 
representation  with  inversion  of  angles. 

Reflection  on  the  jc-axis  determined  by  equations  (4)  or  (5)  is 
a  representation  of  the  same  kind.  If  we  now  combine  trans 
formations  (4)  and  (6)  in  order  to  obtain  the  original  transfor 
mation  (i),  the  two  changes  in  the  sense  of  the  angle,  being 
opposites,  mutually  disappear.  We  can  then  say  — and  it  is  the 
most  important  result  of  this  paragraph : 

IX.  The  transformation  effected  by  z'=i/z  ?'s  a  conformal  repre 
sentation  without  inversion  of  angles. 

EXAMPLES 

1.  Prove  analytically  that  every  circle  which  cuts  the  unit  cir 
cle  orthogonally  is  transformed  into  itself  by  the  transformation 
by  reciprocal  radii. 

2.  Prove  Ex.  i  geometrically. 

*  Konforme  Abbildungw=>  the  term  used  by  GAUSS,  Ges.  Werke,  Vol.  IV,  p.  262, 
and  adopted  universally  by  German  mathematicians.  CAYLEY  used  the  term 
ortfiomorphosis  or  orthomorphic  transformation.  In  general  it  is  the  process  of 
establishing  the  infinitesimal  similarity  of  two  planes  by  means  of  a  functional 
relation  between  the  variables  of  the  planes.  Cf.  also  §  34.  —  S.  E.  R. 


44 


II.    RATIONAL   FUNCTIONS 


3.  Prove  the  converse  of  Ex.  i  for  the  same  transformation. 

4.  Transform  by  reciprocal  radii  a  given  system  of  parallel 
straight  lines.     Do  the  same  for  the  system  of  straight  lines 
orthogonal  to  the  given  system.     Compare  the  two  systems  of 
circles  obtained. 

5.  Transform  by  reciprocal  radii  the  system  of  straight  lines 
through  a  fixed  point  (a  H-  bi).     Discuss  four  cases  according  as 
this  fixed  point  is 

(a)  At  the  origin  ; 

(V)  On  the  unit  circle  ; 

(c]  Inside  of  the  unit  circle  ; 

(d}  Outside  of  the  unit  circle. 

6.  Transform  by  reciprocal  radii  the  system  of  circles  with 
their  centers  at  (a  +  bi)  and  orthogonal  to  the  system  of  Ex.  5, 
discussing  the  same  four  cases  as  in  Ex.  5.     Draw  carefully  the 
accompanying  diagrams  for  both  Exs.  5  and  6. 


HINT.  —  The  lines  through  P  (a  +  bt)  transform  into  circles  through  the 
origin  and  through  /"(the  transform  of  P),  while  the  circles  with  their  centers 
at  (a  +  bi)  are  transformed  into  circles  orthogonal  to  the  first  set. 


§  12.    DIVISION   BY   ZERO  45 

§  12.  Division  by  Zero :  Infinite  Value  of  a  Complex  Variable 
While  addition,  subtraction,  and  multiplication  in  the  field 
of  complex  numbers  are,  as  we  have  seen,  without  exception 
possible  operations,  this  is  not  the  case  with  division.  Among 
the  numbers  introduced  by  us  so  far  there  are  none  which,  when 
multiplied  by  zero,  give  a  definite  number  a  different  from  zero, 
and  hence  none  which  could  be  the  result  of  the  division  indi 
cated  by  a 

o 

as  this  operation  is  defined  in  §  7.  The  function  z'  =  i/z  dis 
cussed  in  the  above  paragraph  is  accordingly  not  defined  for 
2  =  0.  We  may  express  it  otherwise  in  this  way :  when  the 
s-plane  is  represented  conformally  upon  the  z  '-plane,  the  origin 
in  the  z-plane  is  an  exceptional  point  in  the  representation  since 
there  is  no  point  in  the  s'-plane  which  corresponds  to  it. 

But  it  is  customary  in  mathematics  to  remove  such  exceptions 
by  suitable  agreements.  We  make  such  an  agreement  here  in 
the  following  definition : 

I.  /;/  addition  to  the  complex  numbers  and  their  symbols  already 
introduced  we  introduce  now  a  new  one,  "  infinity,'1'1  with  the  symbol 
oo  ,  which  is  to  be  regarded  as  the  result  of  tJie  division  i/o. 

This  analytic  definition  is  parallel  to  the  following  geometrical 
one : 

II.  ///  addition  to  tJie  points  of  the  plane  at  finite  distances  from 
the  origin  let  us  assign  to  tJie  plane  an  infinitely  distant  point  which 
may  be  regarded  as  the  one  corresponding  to  the  origin  in  the  trans 
formation  by  reciprocal  radii. 

It  is  necessary  now  to  determine  the  application  of  these 
terms  and  symbols.  Analytically,  we  state  the  following 
definitions : 


46  II.    RATIONAL   FUNCTIONS 

(1)  III.    a  -\-  oo  =  oo  -\-a  =  oo  , 

(2)  IV.      a  •  oo  =  oo   •  a  =  oo  ,  (a  =£  o), 

from  which  it  is  evident  that  the  fundamental  laws  of  addition 
and  multiplication  spoken  of  in  §§  2  and  3  are  satisfied.  From 
these  we  get  the  following  definitions  for  the  inverse  operations  : 

(3)  oo  —  a  =  oo  , 

(4)  a  —  oo  =  oo  , 
(5) 


^=0' 


00 


(6)  ^  =  oo  ,  (a  *  o). 

a 

The  symbols  oo  ±  oo  ,    o  •  oo,    — ,    — 

O         00 

remain  completely  undetermined,  inasmuch  as  any  number  satis 
fies  the  operations  demanded  by  them.  Thus  it  is  evident  that 
the  desire  to  remove  by  these  definitions  the  exceptions  to  the 
theorems  has  been  very  imperfectly  attained.  For,  while  we 
have  no  additional  case  for  which  one  of  our  operations  would 
be  impossible,  yet  we  now  have  five  indeterminate  forms  instead 
of  the  one,  o/o. 

Geometrically,  we  shall  be  content  for  the  present  to  observe 
that  the  expression  "  A  circle  through  two  points  in  the  finite 
part  of  the  plane  and  the  point  at  infinity  "  signifies  the  same 
thing  as  "  A  straight  line  through  these  two  points."  Thus  Theo 
rem  V,  §  ii  takes  the  following  simple  form :  A  circle  through 
three  points  corresponds  to  the  circle  through  the  three  corresponding 
points  ;  further  details  are  postponed  to  later  paragraphs. 

According  to  conventions  of  this  kind,  certain  words  and 
symbols  previously  defined  are  assigned  a  wider  meaning.  That 
this  procedure  is  permissible  we  have  repeatedly  stated  in  the 
first  chapter  ;  that  it  is  useful  is  justified  by  results.  We  can 
not  object  to  this  on  the  ground  that  in  another  province  of 


§  I3.   TRANSITION   FROM    PLANE  TO   SPHERE  47 

plane  geometry,  the  protective,  another  convention  (infinitely 
many  infinitely  distant  points,  which  lie  on  a  straight  line  at 
infinity)  proves  to  be  of  practical  value ;  we  cannot  expect  that 
one  and  the  same  convention  will  suffice  for  all  our  purposes. 

In  elementary  analysis  the  word  "  infinite  "  is  used  only  in 
theorems  in  a  qualified  sense  (cf.  A.  A.  §  63).  We  speak  here  of 
infinite  as  a  fixed  value.  The  relation  of  these  two  ideas  will 
be  determined  at  our  convenience  (§  31)  after  we  have  fixed 
upon  the  elementary  meaning  of  infinity  as  applied  to  complex 
numbers. 

§  13.   Transition  from  the  Plane  to  the  Sphere  by  Stereographic 

Projection 

Up  to  this  time  we  have  represented  the  complex  numbers  by 
the  points  of  the  plane,  but  in  introducing  this  representation 
(in  §  4)  we  called  attention  to  the  fact  that  any  surface  could  be 
used.  In  particular,  the  sphere  lends  itself  readily  to  this  pur 
pose.  We  therefore  wish  to  apply  to  it  the  representation  of 
complex  numbers  by  the  points  in  the  plane,  which  we  have 
already  introduced.  We  proceed  as  follows  : 

I.  Place  a  sphere  *  of  unit  diameter  on  the  xy-plane  (considered 
horizontal)  so  that  it  touches  the  plane  at  the  origin  O.  The  highest 
point  of  the  sphere  —  that  one  which  lies  diametrically  opposite  to 
O  —  will  be  called  O'.  From  this  point  O',  project  the  points  of  the 
plane  on  the  sphere  by  straight  lines. 

This  kind  of  projection  has  been  used  since  the  earliest  times 
in  cartography  under  the  name  of  stereographic  projection.  Its 
most  important  properties  are  the  following : 

*  This  sphere  is  called  NEUMANN'S  sphere.  In  following  out  one  of  RlEMANN'S 
ideas  Neumann  chose  the  sphere  instead  of  the  plane  as  the  field  of  the  complex 
variable.  It  is  used  by  Neumann  throughout  his  treatise,  Vorlesungen  uber 
Riemann's  Theorie  der  Abelschen  Integrate  (Leipzig,  Teubner,  2d  ed.,  1884). 
—  S.E.R. 


48 


II.    RATIONAL   FUNCTIONS 


II.  To  each  point  of  the  plane  there  corresponds  one  and  only 
one  point  of  the  sphere,  since  each  projector  cuts  the  sphere  in 
only  one  point  besides  the  point  O'. 

III.  Conversely,  to  each  point  of  the  sphere  there  corresponds  one 
point  of  the  plane.     The  point   O'  is  apparently  an  exception  ; 
however,  the  theorem  is  made  general  by  supposing  as  in  pre 
vious  paragraphs  that  the  plane  has  one  infinitely  distant  point 
and  assigning  this  point  to  correspond  to  O'. 

IV.  To  each  straight  line  of  the  plane  there  corresponds  a  circle 
of  the  sphere  passing  through  O' ;  and  conversely. 

V.  Two  such  circles  of  the  sphere  intersect  in  the  same  angle  as 
the  two  corresponding  straight  lines  of  the  plane. 

To    prove  this  theorem,  let  us  pass  the  plane  of  reference, 
Fig.  7,  through  the  vertices  P,  TT  of  both  angles.     The  angle  at 

P  makes  with  O'P 
a  solid  angle.  The 
planes  tangent  to 
the  sphere  at  O 
and  at  TT  cut  this 
solid  angle  in  two 
angles,  the  first 
of  which  is  the 
angle  between  the 


FIG.  7 


straight  lines  of  the  plane,  while  the  second  is  equal  to  the  angle 
between  the  corresponding  circles  on  the  sphere,  since  its  sides 
are  tangents  to  these  circles.  But  both  of  these  planes  are 
normal  to  the  plane  of  reference,  'and  their  intersections  irT, 
PT  make  with  irP  oppositely  equal  angles.  (That  is,  ^  icPO 
',  each  being  complementary  to  the  angle  TrO'O;  and 
^O'OTT,  being  measured  by  half  of  the  same  arc 
O'TT.)  Moreover,  the  two  tangent  planes  with  reference  to  the 
edge  of  the  solid  angle  are  antiparallel  (equally  inclined  to 


§  1 3.    TRANSITION   FROM   PLANE  TO   SPHERE  49 

irP]  *  and  cut  it  accordingly  in  the  same  angle.  Hence,  the  two 
angles  under  comparison  are  equal.  Q.E.D. 

In  this  projection,  points  of  the  plane  indefinitely  near  each 
other  are  transformed  into  points  indefinitely  near  each  other 
on  the  sphere,  and  hence  curves  in  the  plane  tangent  to  each 
other  are  transformed  into  curves  tangent  to  each  other  on 
the  sphere.  Consequently,  the  following  generalization  of 
Theorem  V  is  at  once  possible  : 

VI.  Any  two  curves  of  the  sphere  cut  each  other  at  each  of 
their  points  of  intersection  in  the  same  angle  as  the  corresponding 
curves  of  the  plajie  at  the  corresponding  points  of  intersection. 

We  deduce  further  theorems  with  the  aid  of  analytical  geom 
etry.  We  introduce  the  £,  77,  £  rectangular  space  coordinates  of 
which  the  £-  and  the  7/-axes  coincide  respectively  with  the  x-  and 
the  j'-axes  of  the  (x  +  /v)-plane,  while  the  positive  direction  of 
the  £-axis  is  that  of  OO'.  In  this  system  of  coordinates  the 
equation  of  the  sphere  is 


The  point  (£,  77,  £)  of  the  sphere  corresponds  to  the  point 
of  the  plane  whose  coordinates  are  x,  y  and  radius  vector 
r=  V^+jv2.  To  obtain  the  ^-coordinate  of  this  point  on  the 
sphere  and  its  distance  p  from  the  £-axis,  the  similar  triangles 
<9'<£7r,  7r<f>O,  O'OP  in  Fig.  7  furnish  the  following  double  pro 
portion  :  (  f\*  •  Y  — 

From  this  it  follows  that 

(2}  r=p=^i' 

and  from  these  further  : 

(3)  **-£-.,    I  +  ^  =  -^, 


*  Words  in  the  parenthesis  added  by  the  translator. 


50  II.    RATIONAL   FUNCTIONS 

and  conversely  : 

(4)  C—         ,    •»—          - 


i      r  i      r 

By  construction  it  follows  that 

x:y:r  =  £:rj:P. 
We  find  therefore  that 

VII.  The  coordinates  of  a  point  of  the  sphere  are  expressed  as 
follows  in  terms  of  the  coordinates  of  the  corresponding  point  of  the 
plane  : 

/  %  t          x  y          f          r2- 

(5)  £  =  -    —  ,    iy  =       ,        ,    4  =  ---  -• 

i  -f  r2  i  -f  r2  i  +  r2 

VIII.  Conversely,  the  coordinates  and  radius  vector  of  a  point  of 
the  plane  are  expressed  as  follows  in  terms  of  the  coordinates  of  the 
corresponding  point  of  the  sphere  : 

(6)  XSS-L- 


The  following  theorem  is  obtained  at  once  from  these 
formulas  : 

IX.  To  any  circle  of  the  plane  there  corresponds  a  circle  of  the 
sphere,  and  conversely. 

For,  to  the  points  of  the  plane  satisfying  the  equation  of  the 
circle 

(7)  ar^  +  bx  +  cy  +  d=o, 

there  correspond  the  points  of  the  sphere  whose  coordinates 
satisfy  the  equation 

(8)  ^  +  ^  +  ^-f<i-0  =  o. 

But  this  is  the  equation  of  a  plane  and  it  cuts  the  sphere  in  a 
circle.  This  converse  theorem,  however,  supposes  the  word 
"  circle  "  (in  the  plane)  to  be  taken,  as  in  the  previous  para 
graph,  in  its  extended  sense  to  include  the  straight  line. 


§13.    TRANSITION   FROM    PLANE  TO   SPHERE  51 

We  now  transfer  the  geometrical  representation  of  complex 
numbers  from  the  plane  to  the  sphere  : 

X.  We  assign  to  each  point  of  the  sphere  the  same  complex  num 
ber  z  =  x  +  iy  which  heretofore  belonged  to  its  stereographic  projec 
tion  on  the  plane. 

Thus  to  the  real  numbers  and  the  pure  imaginaries,  for  ex 
ample,  there  correspond  on  the  sphere  the  points  on  the 
"  meridians  "  77  =  o  and  £  =  o  respectively  ;  to  the  points  of  ab 
solute  value  i,  there  correspond  the  points  on  the  "equator" 
£=1/2.  To  opposite  complex  numbers  (IV,  §  2)  correspond 
points  of  the  sphere  which  are  symmetrical  to  the  £-axis,  and  to 
conjugate  complex  numbers  (VIII,  §  4)  correspond  points 
symmetrical  to  the  ££-plane.  To  the  number  oo  introduced  in 
§  12  there  corresponds  on  the  sphere,  just  as  to  any  other  com 
plex  number,  one  and  only  one  point,  viz.  O'. 

By  means  of  this  interpretation  of  complex  numbers  on  the 
sphere  we  can  now  answer  the  question  as  to  what  transforma 
tions  of  the  sphere  (instead  of  the  plane)  into  itself  are  repre 
sented  by  the  functions  heretofore  investigated.  The  functions 
discussed  in  §§  8-10  furnish  nothing  simpler  for  the  sphere  than 

for  the  plane.    However,  it  is  different  with  the  function  z'  =  -  of 

z 

§  ii.  Let  (jc,  y)  and  (x',  y')  be  two  points  of  the  plane  which 
correspond  to  each  other  by  the  transformation  by  reciprocal 
radii  in  reference  to  the  unit  circle  ;  and  let  (£,  77,  £)  and  (£',  r/,  £') 
be  respectively  their  stereographic  projections  on  the  sphere. 
Then  substitute  in  equations  (2),  §  n,  the  values  of  x,  y,  r1  and 
x',  y',  r12  respectively  from  equations  (6)  of  the  present  para 
graph  and  from  the  corresponding  equations  writh  accented 
letters,  and  we  obtain  : 

=-n. 


,-{•{',_{'{'  I- 


52  II.    RATIONAL   FUNCTIONS 

from  these  it  follows  that 

(9)  r~w=*r-i=--«--i)> 

that  is : 

XI.  The  transformation  by  reciprocal  radii  in  reference  to  the 
unit  circle  in  the  plane  corresponds,  by  stereographic  projection  on  the 
sphere,  to  a  reflection  on  the  equatorial  plane  £  —  1/2  =  o. 

The  transformation  of  the  plane  into  itself  by  means  of  z'  =  z~l 
is  performed  by  first  transforming  by  reciprocal  radii  in  refer 
ence  to  the  unit  circle  and  then  reflecting  on  the  axis  of  real 
numbers.  The  corresponding  transformation  of  the  sphere  into 
itself  is  thus  performed  by  two  reflections,  one  on  the  equatorial 
plane  and  the  other  on  the  meridian  plane  t]  =  o.  Now  these 
two  reflections  on  the  two  planes  perpendicular  to  each  other 
are  compounded  by  merely  "  Reflecting  on  the  line  of  intersec 
tion  of  the  two  planes,"  that  is,  by  taking  for  each  point  another 
one  symmetrical  to  the  first  in  reference  to  the  line  of  intersec 
tion.  This  transformation  is  performed  also  by  rotating  the 
sphere  through  180°  about  this  line  of  intersection  as  an  axis. 
Hence,  we  state  the  following  theorem  : 

XII.  The  transformation  z1  =  z~l  determines  a  rotation  of  the 
sphere  through  180°  about  the  diameter  passing  throttgh  the  points 
z  =  i  and  z  =  —  i. 

In  the  plane  the  origin  was  an  exception  to  the  transforma 
tion  in  that  there  was  no  proper  point  corresponding  to  it.  On 
the  sphere,  as  we  have  seen,  it  is  different,  since  the  origin  cor 
responds  to  its  opposite  pole  O'.  Hence  we  say : 

XIII.  The  transformation  z'  =  z~l  is  reversibly  unique  for  all 
points  of  the  sphere  ;  to  any  point  z  there  corresponds  one  and  only 
one  point  2',  and  conversely. 

From  the  geometrical  representation  given  in  XII  we  infer 
further  that : 


§  13-    TRANSITION   FROM    PLANE   TO    SPHERE  53 

XIV.  For  the  transformation  z1  =  z~l  there  are  two  and  only  tu>o 
points  z  which  coincide  with   their  corresponding  points   z{  ',  viz. 
z  =  i  and  z  =—  I. 

We  return  now  to  the  question  postponed  in  a  previous  para 
graph  as  to  how  the  theorems  of  plane  geometry  appear  in  refer 
ence  to  the  convention  introduced  there  ;  it  is  evident  that  this 
convention  amounts  to  regarding  the  plane  as  an  infinitely  large 
sphere  and  transforming  the  theorems  of  ordinary  spherical 
geometry  to  the  plane.  We  shall  not  go  into  further  details  here 
except  to  remark  that  the  circles  of  the  plane  corresponding  to 
great  circles  of  the  sphere  are  then  characterized  by  the  prop 
erty  that  they  cut  the  unit  circle  in  the  end  points  of  a  diameter. 

Further  :  If  we  combine  with  the  transformation  (XII)  that 
reflection  on  the  ^-plane  perpendicular  to  the  diameter,  which 
puts  x  -f-  1  'y  into  —  x  +  iy,  we  find  : 

XV.  The  transformation,  which  replaces  every  point  of  the  sphere 
by  the  one  lying  diametrically  opposite  to  it,  is  expressed  analytically 
by  the  equation  : 

' 


Two  complex  numbers  having  the  relation  to  each  other 
expressed  in  equations  (10)  are  called  diametral. 

EXAMPLES 

1.  The  sphere  may  be  projected  stereographically  upon  a  plane 
as  follows  :  Let  the  center  of  the  sphere  be  taken  as  the  origin 
of  coordinates  £,  77,  £  of  a  point  on  the  sphere.  Let  the  points 
of  the  sphere  be  projected  from  the  south  pole  (whose  co 
ordinates  are  0,0,  —  i)  upon  the  tangent  plane  at  the  north 


54  II.    RATIONAL   FUNCTIONS 

pole  and  take  the  Cartesian  axes  ox  and  oy  on  the  tangent  plane, 
parallel  to  the  axes  £  and  17  respectively.  Show  that  the  co 
ordinates  of  the  projected  point  are 

2J  2r) 

-~ 


f\ 

and  that  x  -f-  iy=  2  tan  -  (cos  <£  +  /  sin  <£),  where  <£  is  the  longi 

tude  measured  from  the  plane  rj  =  o  and  6  the  north  polar  dis 
tance  of  the  point  on  the  sphere. 

2.  A  circle  4  .r2  -f  4  _>'2  —  ^x  —  4y  +  i  =  o  is  projected  upon 
the  sphere  as  in   Ex.  i  ;  find  the  equation  of  the  plane  whose 
intersection  with  the  sphere  represents  this  projection. 

3.  A  circle,  the  equation  of  whose  plane  of  intersection  with 
the  sphere  is  ^  +  ^  +  ^  _  5/4  =  Oj 

is  projected  upon  the  plane  as  in  Ex.   i  ;  find  the  equation  of 
the  projection. 

4.  Solve  Ex.  2  according  to  the  method  of  projection  in  I,  §  13. 

5.  Find  the  equation  of  the  projected  circle  of  Ex.  4  for  the 
plane  and  sphere  as  in  I,  §  13. 

§  14.    The  General  Linear  Fractional  Function  and  the  Circle 
Transformation  * 

In  the  process  of  investigating  rational  functions  of  a  com 
plex  variable  by  proceeding  from  simpler  to  more  complicated 
forms  we  consider  next  the  general  linear  fractional  function  : 


*  The  transformation  determined  by  the  linear  fractional  function,  that  is,  by  the 
linear  substitution,  is  called  bilinear  by  some  authors.  On  Kreisverwandtschaft, 
that  is,  circle  transformation  between  the  planes,  see  MOBIUS:  Abhandlung  der 
Sachs.  Gesellsch.  der  Wissensch.,  1855,  and  earlier  notices  on  the  same  subject  in 
Gesammelte  Werke,  Vol.  II,  p.  243.  —  S.  E.  R. 


§  i4.    THE   GENERAL   LINEAR   FRACTIONAL   FUNCTION    55 

We  take  up  first  the  case 
(2)  ad—bc=o* 

or  a  :  b  =  c  :  d. 

In  this  case  the  transformation  becomes 


all  points  z,  except  z  =  —  bja,  correspond  then  to  the  one  point 
z'  =  a/c,  and  all  points  z',  except  z1  =  a/c,  correspond  to  the  one 
point  2  =  —  b/a.  We  are  thus  dealing  with  a  degenerate  trans 
formation  ;  this  case  is  therefore  excluded  in  what  follows.  We 
shall  discuss  two  additional  cases  : 


I.    Incase  c  =  o, 

z'  reduces  to  the  linear  integral  function  : 


Since  the  discussion  of  this  case  is  already  disposed  of  in  §  10, 
we  omit  it  here. 
II.    In  case 

(5)  '+* 

transformation  (i)  can  be  compounded  from  the  following  three 
simpler  ones  ;  we  first  put 

(6)  *-.+£, 
then 

(7)  '"'  =  77. 
and  finally 


*  That  is,  the  determinant  of  the  transformation  equals  zero.  —  S.  E.  R. 


56  II.    RATIONAL   FUNCTIONS 

The  second  of  these  is  disposed  of  in  §  u,  while  the  first 
and  third  are  similarity  transformations  (§  10).  All  three  of 
these  relations  have  the  property  that  they  transform  circles 
into  circles.  Hence  the  same  property  must  belong  to  the 
transformation  (i)  compounded  from  them.  By  definition, 
therefore : 

III.  Two  planes  having  a  one-to-one  correspondence  are  said  to 
be  circularly  transformed  into  each  other  if  every  circle  of  one  plane 
corresponds  to  a  circle  of  the  other ; 

and  hence  the  theorem : 

IV.  The  z-plane  is  transformed  circularly  into  the  z'-plane  by 
the  linear  fractional  function  (/). 

Since  each  of  the  three  transformations  (6)-(8)  preserves 
angles,  the  same  will  be  true  for  the  transformation  resulting 
from  their  combination.  We  therefore  add  (cf.  VII,  §  n): 

V.  The   representation   is  conformal  without  inversion   of  the 
angle. 

We  shall  call  the  circle  transformation  "  direct "  or  "  in 
verted  "  according  as  the  sense  of  the  angle  remains  the  same 
or  is  changed.  In  the  present  case  we  are  dealing  with  a  direct 
circle  transformation.  An  inverted  circle  transformation  between 
the  z-  and  the  s'-planes  is  obtained  by  putting  z'  equal  to  a  linear 
function  of  the  value  z  conjugate  to  z. 

The  set  of  all  transformations  (i)  possesses  an  important 
property  which  must  be  discussed.  If  in  addition  to  (i)  we 
put  also 

(9)  *"  =  £"' 
it  follows  that 

(10)  z"  =  ^ 


§  14.    THE   GENERAL   LINEAR   FRACTIONAL   FUNCTION    57 

where  the  doubly  accented  coefficients  are  related  as  follows  to 
the  unaccented  and  singly  accented  ones  : 

(i  i)  a"  =aa'  +  cb'  b"  =  ba'  +  db' 

c"  =  ac'  +  ccf  d"  =  be1  +  dd'. 

The  conclusion  therefore  is  that 

VI.  A  linear  function  of  a  linear  function  is  itself  a  linear 
function.     But  it  is  desirable  to  formulate  this  theorem  somewhat 
differently  by  making  use  of  an  important  general  concept  stated 
in  the  following  definition  : 

VII.  A  set  of  transformations  is  called  A  GROUP  when  the  com 
bination  of  any  two  transformations   selected  from    the   set  gives 
always  a  transformation  which  is  itself  contained  in  tJie  set* 

Theorem  VI  therefore  reads  : 

VIII.  The  set  of  all  linear  transformations  forms  a  group. 

(The  special  sets  of  linear  transformations  treated  in  §§  8, 9.  10 
each  form  a  group.  All  these  groups  are  contained  as  "  sub 
groups  "  in  the  group  of  all  linear  transformations.) 

It  is  to  be  noticed  too  that  the  transformation  (i)  can  be 
compounded  in  various  other  ways  from  simpler  transformations. 
(Cf.  the  corresponding  results  of  §  10.)  Considering  the  z- 
and  the  z'  -planes  coincident,  let  us  next  inquire  about  the  fixed 
points  of  transformation  (i),  that  is,  those  points  which  corre- 

*  Present  usage  insists  further  that  the  set  must  also  contain  the  inverse  of  every 
transformation  of  the  set  in  order  that  it  may  be  a  group,  (z'  —  z-\-  a,a  real  and 
>o,  satisfies  the  definition  according  to  VII,  but  is  not  a  group.)  The  inverse 
transformation  is  defined  as  follows :  Given  a  transformation  A  and  suppose  A' 
another  transformation  such  that  when  A  and  A'  are  performed  successively  each 
point  is  transformed  into  itself,  that  is,  the  identical  transformation  is  obtained,  then 
A'  is  called  the  inverse  of  A.  This  is  denoted  symbolically  by  A  •  A~^-=  A~^  •  A  =  I 
where  i  is  the  identical  and  A~l  is  the  inverse  transformation  of  A.  Cf.  Ila,  §  22. 
—  S.E.R. 


58  II.    RATIONAL   FUNCTIONS 

spond  to  themselves  by  this  transformation  and  in  the  determi 
nation  of  which  we  put  z  =  z'.  We  thus  obtain  the  following 
equation  of  the  second  degree  : 

(12)  cz*  +  (d-  a)z-  b  =  o. 

If  the  roots  of  this  equation,  £1}  £2,  are  different,  we  form  the 
following  linear  function  of  z'  : 


According  to  VI  this  is  a  linear  function  of  z  which  can  be  com 
puted.  We  shall  effect  our  purpose  more  directly,  however,  as 
follows  :  For  z=^  we  have  zf=  ^  and  Z  =  o  ;  for  z  =  £2,  ^  =  £2 
and  Z  =  oo  .  But  a  linear  function  of  z  which  becomes  zero  for 
z  =  f  i  and  infinite  for  z  =  £2  must  be  of  the  form 

7  - 

z,  — 


, 
-£2 

where  k  is  a  factor  to  be  determined.     This  may  be  done  by 
noticing  that  for  z  —  o,  z'  =  £/*/*  and  therefore 


hence 


(The  last  form  of  this  result  is  obtained  from  the  fact  that  £t  and 
£2  both  satisfy  equation  (12).)     We  have  thus  found  that 

IX.    If  the  roots  of  equation  (12)  are  unequal,  relation  (i)  be 
tween  z  and  zr  may  be  put  in  the  form  : 

f     \  d  —  £1  _  &  ~  ^£1    z  —  £1 

z'  —  £2      a  —  ^£2    z  —  £2 

*  Any  other  pair  of  corresponding  values  of  z  and  z'  must  naturally  give  the  same 
result ;  thus  for  z  =  —  d/c,  z'  —  <x> , 


§  14.    THE   GENERAL   LINEAR   FRACTIONAL   FUNCTION     59 

But  if  the  roots  of  equation  (12)  are  equal,  both  =  £,  we  form 
the  function 


This  is  then  a  linear  function  of  z  which  is  infinite  for  z  =  £  and 
hence  must  be  of  the  form : 

az  +  ft 

The  coefficients  a,  ft  may  be  determined  from  the  fact  that  the 
equation  :  (az  +  ft-  i)(*  -{)=<>, 

I         Q 

resulting  from  — =  — 

for  zf  =  z,  must  be  identical  with  equation  (12),  and  likewise 
that  £  must  be  a  double  root ;  therefore 

K  +  ft  -  i)  =  o, 

and  Z  takes  the  form  :  — \-  a. 

Here  again  «  is  determined  from  any  two  corresponding  values 
of  z  and  z',  the  simplest  of  which  are  z  =  oo  ,  z'  =  a/c.     Thus 


or,  since  £  = in  this  case, 

2  c 


('7) 


a+d 

*  &  may  also  be  obtained  as  follows  : 


i £ £f  ;  but  this  is  of  the  form  <to  +  P> 


«  +  rf  a  -  c£ 

since  =  —  f  from  (12).     It  follows  that  rt  =  — ^—  =  -^~  .  —  S.  E.  R. 

a^c£      a  +  d 


60  II.    RATIONAL  FUNCTIONS 

Therefore, 

X.    If  the  roots  of  equation  (12)  are  equal,  both  =  £,  the  relation 
(i)  between  z  and  z'  may  be  put  in  the  form 

('«>  -7^--^+    2C 


r-C       z-^a  +  d 

The  equations  (15)  and  (18)  permit  of  a  simple  geometrical 
interpretation.     To  interpret  equation  (15),  put 

£         Z  —   t 

—  =  p(cos  <f>  -f-  i  sm  <i>) 


/£  =  ;;/  (cOS  \\l  +  2  sin  l/f). 

It  therefore  follows  from  equa 
tion  (15)  that 

Now  (cf.  II,  §  7)  p  is  the  ratio  of  the  two  lengths  z£i  and  zt,^ 
<f>  is  the  angle  4V£i-  From  elementary  geometry,  the  geometri 
cal  locus  of  the  points  for  which 

(20)  p  =  const. 


is  a  circle  whose  center  lies  on  the  line  connecting  &  and  £2 
which  has  the  property  that  £1  and  £2  can  be  obtained  from 
each  other  by  the  transformation  by  reciprocal  radii  in  reference 
to  this  circle.  The  locus  of  the  points  for  which 

(21)  <£  =  const. 

is  a  circle  through  £1  and  £2-     Therefore, 

XI.    Transformation  (15)  transforms  each  of  the  two  systems  of 
circles  (20)  and  (21)  into  itself.     All  points  of  a  circle  p  =  a  (or  of 


§  i4.    THE   GENERAL   LINEAR    FRACTIONAL   FUNCTION    6  1 

a  circle  <f>  =  a)  are  transformed  respectively  into  points  of  the  circle 
p'  =  ma  (or  <f>'  =  a  +  ^)  belonging  to  the  same  system. 

Let  us  notice  the  two  special  cases  m  =  i   (that  is,  \k\  =  i) 

and  \l/  =  o  or  =  TT  (that  is,  k  real). 

XII.  In  the  first  of  these  special  cases  each  circle  of  the  first  sys 
tem  is  transformed  into  itself,  and  in  tJie  second  case  each  circle  of  tJie 
second  system  is  transformed  into  itself. 

An  important  property  of  the  two  systems  (20)  and  (21)  is 
that 

XIII.  Every  circle  of  the  one  system   cuts  every  circle  of  the 
other  system  at  right  angles. 

This  is  proved  either  by  elementary  geometry  or  as  follows  : 
The  two  given  systems  are  transformed  respectively  by  the 
linear  transformation 


into  the  system  '  Z  =  const.,  that  is,  into  the  system  of  concentric 
circles  about  the  origin,  and  into  the  system  amZ  =  const.,  that 
is,  into  the  system  of  straight  lines  through  the  origin.  But  both 
of  these  systems  are  orthogonal  to  each  other  ;  and  since  by  (V) 
a  linear  transformation  leaves  angles  unchanged,  the  two  first- 
named  systems  are  also  orthogonal  to  each  other. 

The  special  case  (X)  follows  from  the  general  one  (IX)  by  a 
suitable  limiting  process.  If  we  allow  the  point  &  to  approach 
the  point  £2  in  a  given  direction,  then  the  system  of  circles 
through  £x  and  £2  goes  over  into  the  system  of  circles  through  £2 
and  having  at  this  point  the  given  direction  for  the  direction  of 
the  tangents  ;  the  system  of  circles  which  have  their  centers  on 
the  line  ^£0  and  divide  the  line  segment  £^0  harmonically  are 
transformed  into  the  system  of  circles  which  pass  through  £2  and 
whose  centers  lie  on  the  common  tangent  of  the  circles  of  the 


62  II.    RATIONAL   FUNCTIONS 

first  system,  and  which  have  also  at  £2  a  common  tangent  per 
pendicular  to  the  common  tangent  of  the  first  system. 
Analytically,  this  limit  is  determined  as  follows  :  put 


equation  (15)  now  takes  the  form  : 

(22)  I+^-=(l+« 


Multiply  out,  cancel  i  on  each  side,  divide  by  8,  and  then  let  8 
approach  zero;  equation  (18)  is  the  result.  In  this  process  the 
direction  in  which  £L  approaches  £2  is  left  entirely  undetermined. 
Therefore  (as  an  indirect  result  from  the  first  geometrical  process) 
we  always  obtain  the  same  special  transformation  (18)  from  the 
general  one  (15)  in  whatever  direction  £x  approaches  £2.  It  is 
to  be  noticed  also  that  while  we  found  for  each  such  limit  pro 
cess  just  two  systems  of  circles  which  are  transformed  into 
themselves  by  (18),  there  are  others  having  this  property ;  in 
fact  the  conclusion  is  evident  that  every  system  of  circles 
through  £  having  a  common  tangent  is  transformed  into  itself 
by  (18). 

To  show  this  analytically  let  us  again  put 

-^-  =  Z=jr-MK, 


a  =  (3  +  *y, 
so  that  equation  (18)  reduces  to  the  two  following  ones  : 

(23)  x' 

from  these  it  follows  that 
(24)  (XX1 


§  14.    THE   GENERAL  LINEAR   FRACTIONAL   FUNCTION    63 

that  is,  every  system  of  straight  lines 
(25)  AJf  -h  /A  K=  const- 

is  transformed  into  itself  by  (18).  But,  in  the  Z-plane,  equa 
tion  (25)  represents  a  system  of  parallel  straight  lines  ;  by  the 
transformation  I 

~^  =  Z' 

this  system  is  transformed,  according  to  Va,  §  u,  into  circles 
with  a  common  tangent  at  the  point  £.  Consequently  all  sys 
tems  of  this  kind  are  transformed  into  themselves  by  (18).  It 
follows  therefore  that  : 

XIV.  There  are  infinitely  many  systems  of  circles  each  of  which 
is  transformed  info  itself  by  the  special  transformation  (18)  :    that  is, 
every  system  of  circles  through  £  with  a  common  tangent  has  this 
property. 

However  : 

XV.  Among  these  systems  there  is  o?ie  such  that  any  circle  belong 
ing  to  it  is  transformed  into  itself. 

We  obtain  this  last  system  by  choosing  X  and  p.  in  (24)  so 
that 


EXAMPLES 

1.  Prove  that   the   general    linear   fractional    transformation 
transforms    circles    into    circles    starting   from    the    fact   that 
(z—  a)/(z  —  p)  =  \  is  the  ^-circle  and  then  substituting  for  z  its 
value  in  terms  of  z'. 

2.  What    is    the    condition    that    the    transformation 

,  _  az  +  b 


transforms  the  unit  circle  in  the  s'-plane  into  a  straight  line  ? 

Ans.    \a  =\c\. 


64  II.    RATIONAL   FUNCTIONS 

3.    If  the  invariant  points  for  the  transformation 


cz-\-d 
are  «,  ft  show  that  it  can  be  put  in  the  form 


z'  —  a  __  yZ  —  a 
~~ 


4.    Find  the  invariant  points  for  the  transformation 


i  —  z 
Put  it  in  the  form  given  in  Ex.  3. 

5.  Prove  the  statement  in  the  text  which  says  that  the  geo 
metrical  locus  of  the  points  for  which  p  =  const,  (equation  20, 
§  14)  is  a  circle. 

6.  Discuss  the  transformation  (i)  by  putting  it  in  the  form 

z,_a==_  (ad  -be] 


H) 


Transform  the  origins  in  the  z'-  and  the  ^-planes  into  the 
points  ajc  and  —  djc  respectively.  A  s'-locus  is  therefore  ob 
tained  from  a  s-locus  by  transferring  the  origin  to  —  d/c,  turn 
ing  the  plane  through  two  right  angles  about  the  line  z  = 

j-am  [~za    ] >   inverting  the   locus   in  the  new  position   with 

be  —  ad 


a  constant  of  inversion  equal  to 


,  and  finally  moving 


the  origin  to  the  point  —  a/c. 

7.  Show  by  the  process  in  Ex.  6  that  a  circle  is  transformed 
into  a  circle  by  the  transformation  (i). 

8.  Show  from  the  particular    form   used  in  Ex.  6  that  the 
bilinear  transformation  is  equivalent  to  two  inversions  in  space. 


§  15-    THE   DOUBLE   RATIO   INVARIANT  65 

9.    Show  geometrically   that   the    bilinear   transformation    is 
equivalent  to  two  inversions  in  space. 

[HARKNESS  AND  MORLEY,  Introduction,  etc.  p.  42.] 

10.  Prove  that  the  determinant  of  the  product  of  two  linear 
transformations  equals  the  product  of  the  determinants  of  the 
two  transformations  (understanding  the  product  of  two  trans 
formations  to  be  the  result  of  performing  them  successively). 

§  15.   The  Double  Ratio  Invariant  under  the  Linear  Trans 
formation 

In  §  10  we  saw  that  two  given  s-points  can  be  transformed 
into  two  given  s'-points  by  a  ''  similarity "  transformation 
z'  =  az  -+-  b ;  the  two  constants  a,  b  at  our  disposal  are  deter 
mined  according  to  the  conditions  of  the  problem. 

The  general  type  of  linear  fractional  transformation  (i,  §  14) 
appears  at  first  to  contain  four  arbitrary  constants,  but  there 
are  really  only  three.  For, 

I.  If  u<e  multiply  the  four  coefficients  a,  b,  c,  d  by  tJie  same 
factor  /a,  the  linear  transformation  remains  unchanged ;  it  depends 
therefore  not  upon  four,  but  upon  three  arbitrary  constants  inde 
pendent  of  each  other. 

(If  we  put  m  equal  to  the  reciprocal  of  one  of  the  coefficients, 
unity  takes  the  place  of  this  coefficient,  and  the  formula  appears 
with  only  three  constants  in  it.  But  in  so  doing  we  must  ex 
clude  that  transformation  for  which  this  coefficient  is  equal  to 
zero.) 

It  is  therefore  always  possible  to  determine  the  coefficients 
of  a  linear  transformation  to  satisfy  three  given  conditions.  In 
particular  cases  we  should  investigate  whether  or  not  these 
conditions  are  contradictory  among  themselves.  If,  for  ex 
ample,  it  is  required  to  determine  the  linear  transformation 


66 


II.   RATIONAL   FUNCTIONS 


which  transforms  three  given  distinct  points  %,  zz,  zs  into  three 
other  given  points,  we  have  the  three  equations 


i  -f-  # 


t  '  \ 

,   (*=I,2,  3), 


(0 


or,  fZf 

These  three  equations  are  sufficient  to  determine  the  three 
ratios  of  the  four  coefficients.  As  shown  in  the  theory  of  de 
terminants  (A.  A.  §  31)  it  is  always  possible  to  determine  these 
ratios  for  equations  (i),  and  in  fact  in  only  one  way,  provided 
that  not  all  of  the  four  third  order  determinants  of  the  matrix : 


ZL,       I 


Z2Z2 


are  zero.     But,  according  to  equation  (18),  §  10, 


means  that  the  triangles  (^%s3)  and  (z 
other  ;  and  if 


are  similar  to  each 


=  o      or 


Zi,       I,       I/Zi 


then   the   A  (zjzjzj)  ~  A  f-     -1     i\     If   both   are   true    the 

\Zi       Z2       Z3J 

A  f  i     —     — 

V%        02 


and  therefore 


=  0     or 


=  0, 


§  15-    THE  DOUBLE   RATIO   INVARIANT  6/ 

that  is,  (zl  -  Z2)(z2  -  Z3)(z3  -  zj  =  o  ; 

in  other  words,  two  of  the  z-po'mts  then  coincide.*     The  follow 
ing  theorem  is  therefore  true : 

II.  There  is  always  one  and  only  one  linear  transformation 
which  transforms  three  given  distinct  points  z  into  three  given 
points  z'. 

Of  course,  if  the  transformation  is  not  degenerate  ((2),  §  14), 
the  three  s'-points  must  be  distinct. 

As  an  example  of  theorem  II  we  will  treat  the  problem  to  map 
the  inside  of  the  unit  circle  of  the  z-plane  conformally  upon  that 
half  of  the  s'-plane  whose  points  represent  complex  numbers 
with  the  coefficients  of  /  positive.  For  this  purpose  let  us  asso 
ciate  three  arbitrary  points  of  the  unit  circle  of  the  s-plane  with 
three  arbitrary  real  values  of  zf ;  it  then  follows  that  if,  in 
passing  over  the  series  of  values  %,  z2,  z3  upon  the  unit  circle, 
we  have  the  area  of  this  circle  to  our  left,  then  in  passing  over 
the  corresponding  series  of  values  Zi,  z2,  z3'  upon  the  real  axis 
the  given  half-plane  (called  briefly  the  "  positive  half -plane  ") 
lies  also  to  our  left.  This,  for  example,  is  the  case  when  we 
set  the  points 


respectively  in  correspondence  with  the  points 

,    z3'  =  oo. 

We  thus  obtain  the  equations  : 

,  ^  a-\-  b  ai  +  b  a  —  b 

(2)  — ! —  =  o,    ! —  =  i,    =00, 

c+d  ci  +  d  c-d 

or,  (a  +  3)  =  o,    (b-d)  =  i(c-a),    (c—d)  =  o. 

*  In  this  discussion  z±,  z%,  zs  are  understood  to  be  different  from  zero.  The 
case  where  one  of  these  numbers  =  o  can  be  brought  under  the  general  case  by  an 
auxiliary  transformation  of  some  such  simple  form  as  2'  =  z-\-f. 


68 


II.    RATIONAL   FUNCTIONS 


Since  we  may  put  d=\,  it  follows   that   a  =  —  i,  b  —  i,  c=i, 
that  is, 

(3) 


z'  =  i  • and  hence     z  =  - 


We  investigate  further  the  mapping  of  the  s-plane  upon  the 
plane  by  means  of  these  formulas.     To  the  values 


i, 


correspond  the  values 


oo 


00,          —I,         —  t. 


To  the  s-axis  of  real  numbers  corresponds  the  s'-axis  of  pure 
imaginaries ;  to  the  z-axis  of  pure  imaginaries  corresponds  the 
unit  circle  of  the  s'-plane  (cf.  IV,  §  7).  By  means  of  these 

z'-p/ane 


.    j 

i 

—  -, 

/u 

A 

_J  

~*\ 

o  17 

\ 

IV     1 

^. 

/ 

VII       ^^_ 



^  vw 

-i 

III 


II 


FIG.  9 

given  lines  the  two  planes  are  each  divided  into  eight  regions 
(to  which  the  octants  of  the  sphere  correspond).  These  regions 
correspond  to  each  other  as  in  Fig.  9. 

But  if  four,  instead  of  three,  given  ^-points  are  to  be  trans 
formed  into  four  given  /-points  by  a  linear  substitution,  an 
additional  condition  must  be  satisfied.  This  condition  is  found 
briefly  as  follows :  The  function  of  three  points  (z^—  z<?)/(zz  —  z^ 


§  15-    THE   DOUBLE   RATIO   INVARIANT  69 

already  considered  in  §  10,  equation  (16),  becomes,  by  the  lineal 
transformation  (i), 

z\  —  z'2  _cz3  +  d   Zi  —  z?. 

j       r         r~v  "  ]        ' 

z  3  —  z  2      czl  +  a    z3  —  z2 

that  is,  it  is  in  its  original  form  multiplied  by  a  factor  which 
does  not  contain  z.,.  If  we  now  form  the  quotient  of  this  func 
tion  and  a  corresponding  one  in  which  z4  is  used  instead  of  z2, 
this  factor  disappears.  We  thus  find  : 

/  \ 


The  following  definition  enables  us  to  state  this  result  more 
conveniently  : 

III.  The  double  ratio  of  four  points  (z^  z2,  z3,  z4)  —  taken  in  this 
order  —  is  understood  to  be  the  quotient  : 

(5)  Z^^:*^^=(zl,^z3,z<); 

z3  —  z.2   z2  —  z± 

therefore, 

IV.  The  condition  for  the  existence  of  a  linear  transformation 
which  transforms  four  given  z-points  into  four  given  z'  -points  is,  that 
the  double  ratio  of  the  z  points  shall  be  equal  to  the  double  ratio  of 
the  z1  -points  taken  in  the  same  order. 

And  further  : 

V.  This  condition  is  necessary  and  sufficient  providing  the  four 
given  points  are  distinct. 

For,  when  that  linear  transformation  which  puts  the  points 
%,  z2,  z3  respectively  into  z\,  s'«,  s'3  is  found  by  II,  it  has  the 
property  of  transforming  the  point  z4  into  z'±  providing  the 
double  ratios  (%,  z«,  zs,  z4)  and  (z^,  z.2f,  z3',  z4')  are  equal.  But  there 
is  only  one  such  point  since  equation  (4)  is  of  the  first  degree  in 
2^4.  It  must  therefore  be  the  given  one,  Q.E.D. 


7O  II.   RATIONAL  FUNCTIONS 

The  double  ratio  of  four  complex  points  is  of  course  in  gen 
eral  complex  ;  but  more  precisely  : 

VI.  The  double  ratio  of  four  points  is  real  when,  and  only  when, 
the  four  points  lie  on  a  circle. 

The  amplitude  of  (zv  —  z2)/(z3  —  z2)  is  the  angle  z^z^  and  the 
amplitude  of  (zl  —  z4)/(z3  —  z4)  is  the  angle  ZAZ^.  If  the  quad 
rilateral  z^z2zzz^  is  inscribable  in  a  circle,  then  these  two  angles 
are  inscribed  angles  measured  by  the  same  arc  or  by  arcs  whose 
sum  is  a  whole  circumference.  In  the  first  case  these  angles 
have  the  same  sense,  in  the  second  case  opposite  sense.  More 
over,  in  the  first  case  %.  z3z^  =  ^  z&Zi,  in  the  second  case  it 
=  y.  z&Zt  —  TT.  Therefore  the  amplitude  of  the  double  ratio  is 
zero  in  the  first  case  and  TT  in  the  second,  and  the  double  ratio 
itself  is  real  in  both  cases.  But  if  the  four  points  do  not  lie  on 
a  circle,  then  ^  z3z^  is  different  from  ^  z3z2zlt  and  from  £  z&fa 
—  TT,  and  therefore  the  double  ratio  is  not?  real.* 

If,  in  particular,  zz  =  o,  z3  =  i,  z±  =  oo  ,  we  find  that 

(Zj,  O,  I,  oo  )=;&!, 

that  is : 

VII.  The  double  ratio  of  an  arbitrary  point  z±  with  the  three 
points  O,  I,  oo  is  equal  to  z±  itself. 

As  already  stated,  the  double  ratio  of  four  points  depends  upon 
the  order  in  which  the  points  are  taken.  But  four  points  can 
be  arranged  in  twenty-four  different  ways.  Of  these  the  fol 
lowing  four 

(0!,  Z2,  Z3,  Z4),  (>2,  0!,  Z4,  Z3),  (Z3,  Zi,  Z,,  Z2),   (S4,  Z3,  Z2,  Z^ 

*  To  students  acquainted  with  projective  geometry  we  remark,  without  proving, 
that  the  double  ratio  of  four  points  of  a  circle  as  here  defined  is  exactly  equal  to  the 
double  ratio  of  four  such  points  as  defined  in  projective  geometry  :  the  complex 
double  ratio  defined  here  for  four  given  points  of  the  plane  is  equal  to  their  double 
ratio  upon  that  imaginary  conic  section  determined  by  them  and  one  of  the  "  cir 
cular  points  at  infinity." 


§  15-    THE   DOUBLE   RATIO   INVARIANT  Jt 

give  the  same  double  ratio,  as  a  glance  at  formula  (5)  shows  ; 
thus  only  six  different  double  ratios  can  be  formed  from  the 
same  four  points.  However,  there  are  simple  relations  connect 
ing  these  six  ratios.  If  we  put 

(6)  (X,  32,  23,  24)  =  A, 
it  follows  at  once  that 

(7)  Oi,  Zi,  z3J  Zz)  =  i/A. 

Simple  calculation  shows  further  that 

(8)  fa,  z»  *,,  *4)  =  i-A; 
and,  by  combination  of  these  two  results, 

(9)  (%,  23,  24,  22)  =  i  —  (2X,  34,  23,  2,)  =  i  —  -  =  —  -  —  , 


(\  /  \ 

10)  ($!,  $2,34,23)  = 


A  - 


\**/  \~LJ     "V     ~H     ~d/  \~ 17     -LI     ~<*1     -O/  v 

I   —  A 

It  thus  follows  that : 

VIII.  Each  of  the  six  double  ratios  which  can  be  formed  from 
four  points  is  a  linear  function  of  each  of  the  others. 

The  six  values  (6)-(n)  are  in  general  all  different  from  each 
other.  Two  or  more  of  them  can  be  made  equal  only  for  par 
ticular  values  of  A.  Closer  investigation  shows  that  all  the 
possible  cases  can  be  made  to  depend  upon  the  two  following 
types  by  a  change  of  symbol : 

A          i    _       x       A- i 
A 

and 

A 

(13) 


72  II.    RATIONAL   FUNCTIONS 

IX.  In  case  (12)  we  call  the  four  points  "harmonic"  in  (zj) 
they  are  called  "  equianharmonic , "  * 

For  example,  —  i,  o,  i,  oo  are  four  harmonic  points ;  also  the 
four  vertices  of  a  square ;  the  three  vertices  of  an  equilateral 
triangle  and  the  center  of  its  circumscribed  circle  are  four 
equianharmonic  points  (or  upon  the  sphere,  the  vertices  of  a 
regular  tetraedron).f 

Equation  (4)  may  now  be  expressed  by  means  of  a  term 
which  is  important  in  other  respects.  For  this  purpose  we 
define : 

X.  A  function  of  one  or  more  points  which  remains  unchanged 
when  one  and  the  same  arbitrary  transformation  of  a  given  group 
is  applied  to  all  of  the  points  is  called  an  invariant  of  the  group. 

Thus  equation  (4)  expresses  the  fact  that 

XI.  The  double  ratio  of  four  points  is  an  invariant  of  the  group 
of  linear  transformations.  \ 

We  can  assert  further  that  it  is  the  only  invariant  of  this 
group.  This  is  to  be  understood  as  follows  :  Three  points  can 
have  no  invariant  of  this  group  on  account  of  theorem  II.  The 
equality  of  the  double  ratio  of  two  sets  of  four  points  each  is 
a  sufficient  condition  for  the  existence  of  a  linear  transformation 
which  transforms  the  one  set  into  the  other.  Any  other  func 
tion  of  four  points,  invariant  under  the  linear  transformation, 
must  therefore  have  for  all  sets  the  same  value  for  the  same 
double  ratio.  Hence  it  is  expressed  only  by  this  double  ratio 

*  That  is,  if  A  =  i  the  six  ratios  reduce  to  i,  o,  oo  ;  if  A  =—  i  they  reduce  to  —  i, 
1/2,  2  ;  if  A  =  w  they  reduce  to  <"  or  w2  where  w  is  a  primitive  cube  root  of  unity. 
—  S.E.R. 

f  Also  the  four  points  OPQR  are  harmonic  when  made  by  any  chord  of  a  coni- 
coid  drawn  through  a  point  O  to  intersect  the  surface  in  P  and  Q  and  the  polar  plane 
of  Oin  R.  —  S.E.R. 

J  The  theorem  that  a  double  ratio  is  unchanged  by  a  bilinear  transformation 
was  stated  by  MOBIUS,  Ges.  Werke,  Vol.  II.—  S.  E.  R. 


§  15-    THE   DOUBLE   RATIO   INVARIANT  73 

and  accordingly  is  not  counted  as  a  new  invariant.  But  there  is 
no  new  invariant  for  more  than  four  points.  That  is,  suppose 
F(zl,  *2,  '"  zn)  to  be  a  function  of  n  (^  4)  points  invariant  under 
the  group  of  linear  transformations.  In  place  of  n  —  3  points, 
z4,  £5,  •••  zn,  let  us  put  the  n  —  3  double  ratios  which  are  formed 
by  the  remaining  three  points,  zt,  z,,  z3,  with  each  of  these  n  —  3 
points.  F  is  then  a  function  of  the  n  —  3  double  ratios  and  of 
21?  ^2,  z3.  If  now  it  is  an  invariant,  it  must  take  on  the  same 
value  for  pairs  of  sets  of  n  points :  z±,  z2,~-  zn  and  z±,  z2f,  •••  zn' 
which  are  set  in  correspondence  by  the  linear  transformation 
(i).  But  since  the  ;/  —  3  double  ratios  take  on  the  same  value 
for  even-  pair  of  sets,  either  a  relation  between  01}  z2,  z3  and  0/, 
%',  z3f  must  remain  or  F  must  be  a  function  of  the  n  —  3  double 
ratios  alone.  The  first  is  impossible  on  account  of  theorem  II, 
and  hence  /MS  expressed  by  the  «  —  3  double  ratios. 

EXAMPLES 

1.  What  is   the    most  general    algebraic   relation   between  z 
and  z1  which   gives  a    one-to-one    correspondence  between    the 
points  of  the  z-  and  the  s'-planes  ? 

2.  Determine     the    linear    fractional    transformation    which 
puts  the  points  z  =  —  i,  o,  2   respectively  into  the  points  z'  =  o, 

I,    00. 

3.  Determine  as  in  Ex.  2  the  relation  which  transforms  i,  /,  3 
respectively  into  o,   i,   oo. 

4.  What  relation  between  z  and  z'  will  transform  the  cube 
roots  of  unity  i,  oo,  to2  respectively  into  o,  i,  oo  ? 

5.  Where  is  the  point  z'  corresponding  to  z  =  —  djc  by  the 
transformation  z'  =  (az  +  b)j(cz  -f-  //)  ? 

6.  Let  c'  =  (2  2+3)7(3  z  —  2).     Show  that  the  center  of  the 
^-circle  passing  through  the  points  corresponding,  by  this  trans- 


74  II-    RATIONAL   FUNCTIONS 

formation,  to  the  points  z  =  o,  z,  —  i  is  at  the  point  z1  =  —  -f% 
and  its  radius  is  if.  Find  also  the  center  and  radius  of  the 
^'-circle  corresponding  to  the  points  z  =  o,  2  /,  —2*;  also  to 

z  =  /,   —  2  ,  2  z  . 

7.  Determine  the    function   z'  =f(z)   which   maps  the   recti 
linear  triangle  whose  vertices  are  z  =  o,  z  =  i,  2  =  i  +  i  on  the 
half-plane,  these  three  points  going  over  respectively  into  the 
points  z'  =  oo,  z'  =  o,  z'  =  i.     To  which  half-plane  does  this  tri 
angle  correspond  ? 

8.  Determine    the    linear    fractional    transformation    which 
transforms  the  points  z  =  i,  z  =  —  i,  z  —  i  respectively  into  the 
points  z'  =  2,  z'  =  o,  z1  —  oo. 

9.  A  circle  of  radius  r  and  center  (h,  k)  in  the  2-plane  is 
transformed  into  a  circle  in  the  s'-plane  by  the  substitution 


show  that  the  radius  of  the  new  circle  is 

r  ad—  be 
\        c2 

where  A  =  (p  cos  0  +  hj-  +  (p  sin  (9  +  Kf  —  rz  and  /o,  6  are  the 
modulus  and  the  amplitude  respectively  of  djc.  Find  also  the 
coordinates  of  the  center  of  this  new  circle. 

The  equation  of  a  circle  whose  center  is  at  (h,  k)  and  radius 
r  can  be  put  in  the  form  (z  —  h  —  ki)(z  —  h-\-  ki]  =  rz  or  zz  +  Az 
+  A  •  z  +  y  =  o  where  A  =  —  h  +  ki,  A  =  —  h  —  ki  and  y  =  A  •  A 
—  r1  and  dashes  indicate  conjugate  imaginaries.  This  equation, 
conversely,  represents  a  circle  when  A,  A  are  conjugate  imagi 

naries  and  y  is  real.     Its  center  is     —  *  —  —  —  ~,  />  -  '    and 

>•  .'.        2    J 

its  radius  is  (AA  —  y)-.  Now  subject  this  circle  to  the  trans- 
formation  *<  =  (az  +  b)  /  (cz  +  <t) 


§  15.    THE   DOUBLE   RATIO   INVARIANT 


or,          0  =  (-  dz'  +  ti)l(cz*  -  a)  and  z  =  (-  afz'  +  b)/(c-  a) 
and  we  get  the  relation 

S'0'0'  +  A'0'  +  A'?  +  y'  =  o. 

Determine  these  coefficients  8f,  A'  A',  and  y'  and  show  that  A',  A1 
are  conjugate  imaginaries  and  that  B1.  y'  are  real.  It  therefore 
represents  a  circle  whose  center  and  radius  can  be  determined. 

10.  Divide   the  0-plane  into  eight  regions  by  means  of   the 
axes  and  the  unit  circle.     Find  the  regions  in  the  0'-plane  which 
correspond  by  the  transformation  z'  =  (i  4-  z)/(i  —  z)  to  each  of 
these  regions.     Is  this    transformation    involutoric  ?     Compare 
the  unit  circle  and  the  axis  of  imaginaries. 

11.  In  VI,  §  15  it  is  shown  that  four  points  lie  upon  a  circle 
when  and  only  when  their  double  ratio  is  real.     Another  form 
of  this  condition  is  that  it  is  possible  to  choose  real  quantities 
<z,  b,  c  such  that 


i        ,  i        ,  i 

a       ,  b       ,  c 


=  0. 


Observe  that  the  transformation  0'=i/(0  — 04)  is  equivalent 
to  an  inversion  with  respect  to  the  point  04  together  with  a  cer 
tain  reflection.  If  015  02,  03  lie  on  a  circle  through  04  the  cor 
responding  points  Zi  =  i/(zl  —  04),  s.2'  =  i/(z.2  —  04),  03'=  i/(03— 04) 
lie  on  a  straight  line.  Hence,  by  Ex.  23,  Chap.  I,  we  can  find 
real  quantities  a',  V,  c'  such  that  a1  +  b'  +  c'  =  o  and 

a'  b1  c' 

-H h-     -  =  o, 

0!  -  04         Z,  -  04         03  -  04 

and  it  follows  easily  that  this  is  the  given  condition. 

12.  The  set  of  all  linear  fractional  transformations  forms  a 
group,  since  the  compound  of  any  two  of  them  is  again  one  of 


/6  II.    RATIONAL   FUNCTIONS 

the  same  kind.  What  is  the  relation  between  this  group  and 
the  set  of  all  the  transformations  represented  by  z'  =  z  +  (3 
where  (3  has  all  positive  and  negative  values?  Discuss  in  the 
same  way  z'  =  az,  and  z1  =  az  -\-  ft. 

13.  Find  the  six  double  ratios  of  the  points  o,  i,  oo,  z. 

14.  If  the  double  ratio  of    z,  %,  z2,  z3  =  —  <o,  find  z  (o>  is  a 
primitive  cube  root  of  unity). 

15.  Prove  the  theorem  that  in  inversion  in  space  lengths  of 
double  ratios  are  preserved,  that  is,  that  the  length  (A,  B,  C,  £>)  or 

is  equal  to  the  length  (A',  B',  C,  D')  or 


Invert  the  points  with  reference  to  a  sphere.  Since  OA  -  OA' 
=  OB  •  OB',  the  triangles  are  similar  and  hence  OA  :  OB  :  AB 
=  OB1  :  OA' :  A'B'.  Therefore 

OA'  •  OB' 


OA 


Similarly  for  A ] D\  C'B',  and  CJ>\  then  substitute  these 
values  in  the  expression  for  the  length  (A1,  B' ,  C,  D'),  reducing 
finally  to  the  expression  for  the  length  (A,  B,  C,  D}. 


§  16.    LINEAR   TRANSFORMATION   ON  THE   SPHERE        // 

16.  Prove  that  any  rotation  of  NEUMANN'S  sphere  about  any 
diameter  as  an  axis  corresponds  to  a  linear  fractional  trarfsfor- 
mation  in  the  plane  tangent  at  the  origin. 

Consider  four  points  projected  stereographically  before  and 
after  the  rotation  ;  consider  also  the  double  ratio  of  these  four 
points,  using  the  theorem  of  Ex.  15.  Let  the  points  a,  b,  c,  z  pro 
ject  into  a\  V,  e',  z'  and  A,  B,  C,  Z  into  A\  B' ,  C,  Z' ;  at  the 
conclusion  solve  for  z'  as  a  linear  fractional  function  of  z. 

17.  In  IX,  §  15,  the  double  ratio  of  four  points  %,  z2>  z3,  z4  is 
called  "  harmonic  "  when  it  is  equal  to  —  i.     Show  that  in  this 
case    2/(zi  —  z3)  =  i/(zi  —  z2)  +  i/(zi  —  z4).      Why    is    it   called 
"  harmonic  "  ? 

§  16.     Significance  of  the  Linear  Transformation  on  the  Sphere ; 
Collineations  of  Space  Corresponding  to  It 

We  will  interpret  the  results  of  §  14  further  by  stereographic 
projection  on  the  sphere.  The  circles  of  the  plane  which  pass 
through  the  points  £1?  £2  correspond  to  the  circles  through  the 
corresponding  points  on  the  sphere,  or  otherwise  expressed : 
they  correspond  to  the  curves  of  intersection  made  by  the  planes 
of  a  sheaf  of  planes  whose  axis  cuts  the  sphere  in  these  two 
points.  But  there  are  also  circles  on  the  sphere  cut  out  by  a 
sheaf  of  planes  that  correspond  to  the  system  of  circles  repre 
sented  by  equation  (20),  §  14  (the  difference  being  merely  that 
in  this  case  the  axis  of  the  sheaf  does  not  cut  the  sphere).  This 
is  evident  from  the  following : 

Let  us  draw  planes  tangent  to  the  sphere  at  all  points  of  a 
circle  of  the  sphere ;  they  thus  envelop  a  right  circular  cone ; 
the  vertex  of  this  cone  is  called  the  pole  of  the  plane  of  this 
circle  with  respect  to  the  sphere.  Any  element  of  the  cone  is 
at  right  angles  to  the  tangent  to  this  circle  at  its  point  of  contact 
nd  thus  coincides  with  the  tangent  to  those  circles  of  the 


78  II.    RATIONAL   FUNCTIONS 

sphere  which  cut  the  first  at  right  angles  at  this  point.  Thus 
the  plane  of  each  such  circle  must  contain  this  element  of  the 
cone  and  thus,  too,  the  vertex  of  the  cone,  the  pole  of  the  first 
circle.  The  plane  of  every  circle  which  cuts  two  given  circles 
of  the  sphere  at  right  angles  contains  accordingly  the  poles 
of  the  planes  of  both  circles  and  thus,  too,  the  line  connecting 
them.  The  proof  thus  follows  in  consideration  of  XIII,  §  14. 
Hence  we  may  say  : 

I.  Every  linear  transformation  whose  fixed  points  are  distinct 
transforms  into  themselves  two  systems  of  circles  on  the  sphere,  each 
of  which  results  from  the  intersection  of  a  sheaf  of  planes  with  the 


We  can  now  think  of  a  definite  transformation  of  space  into 
itself  as  corresponding  to  every  such  transformation  of  the 
sphere  into  itself,  by  which  every  plane  which  intersects  the 
sphere  (of  course  in  a  circle)  is  transformed  into  another  plane 
which  intersects  the  sphere  in  the  circle  corresponding  to  the 
first.  Since  all  the  circles  through  two  points  %,  z2  correspond 
to  the  circles  through  the  corresponding  points  z± ,  j&2',  it  follows 
that :  to  all  the  planes  which  intersect  in  a  straight  line  cutting 
the  sphere,  correspond  the  planes  of  a  second  such  sheaf. 
Further,  since  all  the  circles  which  intersect  two  given  circles  at 
right  angles  correspond  to  circles  which  intersect  the  two  corre 
sponding  circles  at  right  angles  (from  V,  §  14),  it  follows  also 
as  was  just  proved  that :  to  all  the  planes  of  a  sheaf  whose  axis 
does  not  intersect  the  sphere,  correspond  the  planes  of  a  second 
such  sheaf.  In  this  way,  therefore,  all  the  straight  lines  in 
space  are  arranged  in  pairs.  And  since  the  theorem  holds  that, 
when  several  straight  lines  not  all  in  the  same  plane  are  arranged 
in  pairs,  they  all  go  through  the  same  point,  it  follows  that  all 
straight  lines  through  a  point  correspond  again  to  straight  lines 


§  16.   LINEAR  TRANSFORMATION   ON  THE   SPHERE       79 

through  a  point.  By  this  transformation  the  planes  of  space, 
and  thus  the  points  of  space  as  well,  are  set  in  a  correspondence 
reversibly  unique.  A  transformation  of  this  kind  is  called  a 
collineation  ;  accordingly,  we  can  write  : 

II.  To  each  linear  transformation  of  the  complex  variable  z  on 
the  sphere  there  corresponds  a  collineation  of  space,  which  trans 
forms  the  points  of  the  sphere  precisely  in  the  same  way. 

According  to  theorem  I,  this  collineation  in  the  general  case 
belongs  to  that  particular  kind  which  transforms  into  themselves 
two  straight  lines,  two  real  points  of  one  of  these  straight  lines 
(viz.  its  points  of  intersection  with  the  sphere),  and  two  real 
planes  through  the  other  straight  line  (the  planes  through  it  tan 
gent  to  the  sphere).  In  the  special  case  (XIV,  XV,  §  14)  a  real 
point  of  the  sphere,  each  tangent  at  this  point  and  each  plane 
through  a  definite  one  of  these  tangents,  is  transformed  into  itself. 

On  the  basis  of  the  formulas  of  §§  13  and  14  it  would  not  be 
difficult  (even  though  cumbersome),  to  carry  out  this  process 
analytically  and  thus  to  find  the  equations  of  the  corresponding 
collineation  for  each  linear  transformation  of  z.  We  will  do  this 
only  for  that  transformation  which  corresponds  to  a  translation 
in  the  plane  parallel  to  the  jc-axis.  For  this 

d  =  z  -f-  «  («  is  real),      that  is,      x'  =  x  +  a,  _/  =  y. 

Accordingly  we  have  the  following  results  from  the  formulas  (6), 
§  13,  and  those  which  are  obtained  from  (5)  by  accenting  all  the 
letters : 


(0 


2  ax 

y  y 


-f  /-2+  2  ax  +  a2      2  a£+(i  H-a2)(i  — 
rz+2  ax  +  «2  2  «£  +  «2i  — 


80  II.    RATIONAL   FUNCTIONS 

(There  are  of  course  an  infinite  number  of  transformations  of 
space  which  transform  the  points  of  the  sphere  as  desired. 
The  process  shows  that  we  obtain  the  required  collineation  if 
we  do  not  make  explicit  use  of  the  equations  of  the  sphere, 
but  use  the  formulas  of  §  13  exactly  as  found  there.) 

A  particular  case  of  collineation  is  found  in  the  "  Motions  in 
Space,"  that  is,  in  those  transformations  which  transform  each 
figure  into  one  congruent  to  it.  If  we  take  for  granted  at  the 
outset  that  any  movement  which  puts  a  sphere  into  itself  can  be 
replaced,  so  far  as  the  result  is  concerned,  by  a  rotation  of  the 
sphere  on  a  diameter  as  an  axis,  we  can  then  easily  determine 
all  such  movements  and  the  linear  transformations  correspond 
ing  to  them.  To  this  end  we  return  to  equation  (15),  §  14.  If 
this  is  to  represent  a  rotation  of  the  sphere  about  a  diameter  as 
an  axis  then  first,  &  and  £2  must  be  diametral  points  (XV,  §  13) ; 
and  second,  if  each  of  the  circles  p  =  const,  from  equation  (20), 
§  14,  which  in  this  case  are  parallel  circles,  are  to  be  trans 
formed  into  themselves,  it  follows  that  m  must  =  i,  that  is,  k 
must  be  an  expression  of  absolute  value  i.  Hence  if  a  and  A 
are  quantities  of  absolute  value  i  and  r  a  positive  real  number, 
we  can  put 

£j  =  ra,      £2  =  —  r~^-&,      and  k  =  A2. 

The  solution  of  equation  (15),  §  14  for  z'  thus  takes  the  form : 

^  =  z(ra_+  r~W)  +  «2(*  ~  *•*} . 
z(  I  -  A2)  +  (r-^a  +  r«A2)  "  ' 

or,  by  multiplying  numerator  and  denominator  by  a~l\~l : 
z(rX~l  -f  r~l\)  +  a(X.~l  -  A) 


za~\\.~l  -  A)  -f-  (r^A-1  +  rA) 

Here  the  coefficients,  apart  from  the  sign  of  one  of  them,  are 
conjugate  to  each  other  in  pairs  (for  A"1  is  conjugate  to  A,  or1 


§  16.   LINEAR  TRANSFORMATION   ON   THE   SPHERE       8  1 

to  a,  and  r  is  real)  ;  if  A,  B,  C,  D  are  real  numbers,  we  can 
therefore  write  : 


III.  Therefore  a  linear  transformation  of  z  can  always  be  put  in 
the  general  form  (2)  when  it  represents  a  rotation  of  the  sphere 
about  its  center* 

*  EULER'S  representation  of  rotation  about  a  fixed  point  is  obtained  from  equa 
tion  (2)  of  the  text  by  introducing  the  space  coordinates  £,  7?,  £,  and  f,  TJ',  £'  by 
means  of  formulas  (5)  and  (6)  of  §  13.  We  thus  obtain  : 

x'+iy' 


(Or-  Dy  +  A)  +  i(Dx  +  Cy-B) 

AD)y  + 


(6-2  +  Z>2)/*  +  a(AC-  BD)x  +  z(-AD  -  BC)y  +  A*  +  &  ' 


(—  ^Z)  -  BC)y+  (.4*  +  ^2) 
If  we  put 


C^  4- 

it  follows  that 


The  numerator  on  the  right-hand  side  is  : 
[(AC+  BD)  4-  i(BC-  ADftr*  +  [(A*  -  B*  - 
CD)  +  /(^-  - 


Now  introduce  the  coordinates  ^,  ij,  ^  and  we  obtain  : 

i(BC— 


+  [2(  -  AB  +  CY>)  +  i(A*  -B*+&—  Z>2)]  r,, 
and  by  dividing  into  real  and  imaginary  parts  : 


(6)      ^'  =  z(AB+  CD)S+(A*—B*+C*—I*)i  +  3(BC-'j4D)({—  1/2). 
And  from  (a)  it  then  follows  that 

I  -  r'2  =  (^2-|-^2—  C2—  ^(t  —  ra)  +  4(.^c—  BD)x  — 


82  II.    RATIONAL   FUNCTIONS 

§  17.    The  Function  z1 

In  the  preceding  paragraphs  we  investigated  in  detail  linear 
functions  of  z.  We  now  turn  our  attention  to  the  function 

(1)  W=  Z  •  Z  =  Z2. 

We  express  w  and  z  first  in  rectangular  and  then  in  polar  coor 
dinates  ;  accordingly : 

(2)  z  =  x  -f-  iy  =  ^(cos  cf>-\-  i  sin  <£), 

(3)  w  =  u  +  iv  —  p(cos  i/'  +  z  shu//), 

and  therefore  from  (n),  §  3,  we  obtain : 

(4)  u  =  x*  —  y2,    v=2  xy, 
and  from  (i),  §  6, 

(5)  p  =  r\    «A  =  2</>. 

The   formulas  (4)  determine    one   and   only    one    pair  of   real 
values  (u,  v)  for  each  pair  of  real  values  (x,  y) ;  we  say : 

I.  The  function  w  =  zz  is  hence  said  to  be  single-valued  over 
the  entire  plane. 

The  construction  of  a  point  w  corresponding  to  a  definite 
point  z  is  most  conveniently  obtained  by  using  formulas  (5) ; 
the  radius  vector  of  such  a  point  w  is  to  the  radius  vector  of  z 
as  that  of  z  is  to  unity,  while  the  amplitude  of  w  is  double  the 
amplitude  of  z. 

To  each  circle  (r—  const.)  about  the  origin  of  the  2-plane 
corresponds  a  circle  (p  =  const.)  about  the  origin  of  the  o/-plane. 

and  then 

(e)     N(^'-I/2)=-2(AC-SD^  +  2(AD+BC)r]-i-(A^  +  B^-C^-D^^-I/2). 

The  formulas  (£)-(«)  are  precisely  those  due  to  EULER. 

It  is  sufficient  to  say  without  proving  that  every  linear  transformation  oi  x -\- iy 
determines  a  movement  in  space  considered  not  from  the  standpoint  of  Euclidean 
geometry  but  from  that  non-Euclidean  geometry  for  which  the  sphere  is  the  fun 
damental  surface. 


§  1  7.  THE   FUNCTION  z2  83 

If  the  radius  of  the  first  circle  increases  continuously  from  o  to 
oo,  then  the  radius  of  the  ay-circle  takes  on  all  values  continu 
ously  increasing  from  o  to  oo  (as  is  known  from  the  real  func 
tion  r"1  of  the  real  variable  r,  A.  A.  §§  46,  61).  To  each  straight 
line  <£  =  const,  through  the  origin  of  the  2-plane,  there  corre 
sponds  a  straight  line  ty  =  const,  through  the  origin  of  the  a/ 
plane.  But  the  amplitude  of  the  latter  line  (on  account  of  the 
second  one  of  equations  (5))  takes  on  all  values  from  o  to  2  TT 
continuously,  while  the  amplitude  of  the  first  takes  on  only  the 
values  from  o  to  TT.  These  two  results  are  stated  in  the  follow 
ing  theorem  : 

II.  The  positive  half  of  the  z-plane  (that  is,  that  part  of  the 
plane  which  includes  the  points  z  —  x  +  iy  where  y  is  positive)  is 
mapped  continuously  and  uniquely  upon  the  w-plane  by  means  of 
the  function  w  =  zz. 

And  this  mapping  is  reversely  unique.  For,  p  =  rz  and  \f/  = 
2  <f>  take  on  each  of  the  above  pair  of  values,  p  between  o  and 
+  cc,  ^  between  o  and  2  TT,  only  once  while  r  increases  from  o 
to  +  oo  and  <£  from  o  to  ?r.  On  the  contrary,  the  continuity  in 

w-p/ane  z-p/a/ie 


\\\\\\\\\\\\\\\AAAAAAA 


FIG.  10 

this  case  is  interrupted  along  the  positive  half  of  the  real  axis 
of  the  w-plane,  inasmuch  as  the  two  sides  of  this  positive  half- 
axis  correspond  to  the  positive  and  negative  parts  of  the  real 
axis  in  the  positive  half  of  the  s-plane  as  indicated  in  Fig.  10. 

If  <£  increases  further  from  TT  to  2  TT,  then  \j/  takes  on  the 
values  from  2  TT  to  4  TT  ;  that  is,  the  ray  ^  =  const,  sweeps  over 
the  whole  plane  again  so  that  the  negative  s-half-plane  is  also 


84  II.    RATIONAL   FUNCTIONS 

mapped  continuously  and  uniquely  upon    the  w-plane.     From 
this  we  therefore  conclude  that : 

III.  The  function  w  =  z*  fakes  on  each  complex  value  w  differ 
ent  from  o  and  oo,  in  two  and  only  two  points  of  the  z-plane. 

Moreover,  two  such  points  are  connected  by  the  relation 
£2  =  —  zv ;  this  follows  easily  from  the  left  side  of  the  equation 
z?  —  z?  =  o,  the  factors  of  which  are  %  —  %  and  z2  +  %.  We  are 
interested  in  this  relation  particularly  because  it  is  linear;  we 
define  as  follows : 

IV.  A  function  w=f(z)  which    remains    unaltered  when   we 
substitute  in  it  a  definite  linear  function  of  z  in  place  of  z  is  called 
a  function  with  a  linear  transformation  into  itself  or  an  automor- 
phic  function* 

Part  of  theorem  III  may  thus  be  stated  more  precisely : 

V.  The  function  w  —  z2  is  an  automorphic  function.     It  remains 
unchanged  when   subjected  to    the    linear    transformation   of  the 
variable : 

(6)  «••,_* 

Further,  let  us  now  introduce  the  following  definition : 

VI.  A  region  in  which  a  single-valued  function  w  of  z  takes  on 
all  of  its  values  once  and  only  once  is  called  a  t  fundamental  region 
for  this  function. 

It  therefore  follows  from  the  definition  of  an  automorphic 
function  and  of  a  fundamental  region  that : 

VII.  If  a  fundamental  region  of  an  automorphic  function  is 
known  and  if  it  is  mapped  on  a  second  region  by  one  of  the  trans 
formations  of  the  function  into  itself,  then  this  second  region  can 

*  A  special  kind  of  automorphic  functions  are  the  periodic  functions.     Cf.  §  41 . 
also  Ex.  4  at  the  end  of  §  18,  and  Ex.  31  at  the  end  of  Chap.  IV.  —  S.  E.  R. 
f  Not  however  "  the." 


§  1 7.   THE   FUNCTION  z2  85 

tiow here  overlap  the  first ;  it  is  also  a  fundamental  region  of  the 
automorphic  function. 

Thus  each  of  the  two  half-planes  separated  by  the  axis  of 
real  numbers  are  fundamental  regions  for  the  function  z*. 

We  shall  continue  somewhat  in  detail  the  mapping  of  the 
2-plane  on  the  «'-plane  by  the  function  w  =  z2.  For  this  pur 
pose  let  us  determine  what  curves  of  the  w-plane  correspond 
to  the  lines  parallel  to  the  axes  of  the  0-plane.  If  we  put  y  =  c, 
equations  (4)  express  //  and  v  in  terms  of  the  auxiliary  variable 
x,  the  elimination  of  which  gives 

(7) 


For  every  definite  value  of  c  this  is  the  equation  of  a  parabola 
which  has  the  //-axis  for  major  axis  and  the  line  u  =  —  c*-  for  the 
tangent  at  the  vertex.  Putting  the  equation  in  the  form 

(8)  u-  +  v-  =  (u  -h  2  c-)\ 

we  see  that  the  origin  is  the  focus  and  the  line  u  4-  2  c1  =  o  is 
the  directrix.  Since  c  is  essentially  real  and  c2  therefore  posi 
tive,  the  directrix  crosses  the  negative  half  of  the  #-axis,  and 
the  parabola  stretches  to  infinity  toward  the  right.  The  focus 
and  the  major  axis  are  independent  of  c.  Parabolas  with  the 
same  focus  and  the  same  major  axis  are  called  confocal.  We 
put  these  results  in  the  following  form : 

VIII.  The  straight  lines  of  the  z-plane  parallel  to  the  x-axis  are 
transformed  by  the  function  w  =  z-  into  a  system  of  confocal  parab 
olas  which  Jiave  the  origin  for  focus  and  the  u-axis  for  major  axis 
ami  which  open  in  the  direction  of  positive  u. 

Moreover,  if  we  put  x  =  c  in  equations  (4)  and  eliminate  y. 
we  obtain : 

(9)  ««-.)  =  ^Y, 


86  II.   RATIONAL   FUNCTIONS 

or, 

(10)  *2  +  ^  =  (*-2^)2, 

that  is : 

IX.  The  parallels  to  the  y-axis  are  transformed  into  parabolas 
which  have  the  same  focus  and  the  same  major  axis  as  those  in 

VIII,  but  which  open  in  the  direction  of  negative  u. 

In  general,  it  can  be  shown  that  any  straight  line  of  the  z- 
plane  which  does  not  go  through  the  origin  is  transformed  into 
a  parabola  of  the  w-plane  which  has  the  origin  for  focus. 

The  converse  question :  What  curves  of  the  z-plane  map  into 
straight  lines  of  the  w-plane  ?  —  will  be  answered  as  follows : 
Let  the  equation  of  such  a  straight  line  be 

(u)  au -\-bv-\-c-Q\ 

replace  u  and  v  in  this  equation  by  their  values  from  (4) ;  we 

thus  obtain : 

(12)  a(x*  —  y2)  4-  2  bxy  +  <r  =  o. 

X.  This  is  the  equation  of  a  conic  section,  and  in  fact  an  equi 
lateral  hyperbola  (since  the  coefficients  of  x1  and  jy2  are  equal 
but  opposite    in  sign)  whose  center  is  at  the  origin  (since    the 
terms  of  first  degree  in  x  and  y  are  absent). 

Parallel  straight  lines  (whose  equations  differ  only  in  the  value 
of  c)  thus  correspond  to  hyperbolas  with  the  same  asymptotes. 
Parallels  to  the  z^-axis  (?/-axis)  correspond  to  hyperbolas  which 
are  asymptotic  to  the  coordinate  axes  (to  the  bisectors  of  the 
angle  between  the  coordinate  axes,  resp.). 

It  is  important  also  to  notice  that  the  map  determined  by  the 
function  w  =  s?  is  conformal  (VII,  §  n).  We  shall  prove  this 
most  easily  by  using  the  equations  (5).  If 

#-/(r) 

is  the  equation  of  a  curve  in  the  s-plane  in  polar  coordinates, 


§  1 8.   THE   FUNCTION  w  =  zn  87 

then  the  tangent  of  the  angle  between  the  curve  and  the  radius 
vector  is 

r& 

dr' 

For  the  corresponding  curve  of  the  7£'-plane  we  obtain 

f     x  d\b        o    2  f/<f>  d<$> 

(H)  n—L-  =  r-- —  =  r  •  — -, 

P4>  -rdr  dr 

from  equations  (5).  Thus  the  two  angles  are  equal  to  each 
other*  ;  we  conclude  from  this,  as  in  VI,  §  n,  that  the  angles 
between  any  two  corresponding  curves  are  equal  to  each  other. 
We  say : 

XI.  The  function  w  =  z1,  just  as  the  linear  functions  investi 
gated  in  §§  8-16-,  determines  a  conformed  representation  without 
inversion  of  the  angle. 

However,  there  is  one  exception  to  be  made.  Equation  (13) 
proves  nothing  for  the  corresponding  origins  of  the  two  planes, 
since  the  expressions  lose  their  meaning  at  these  points.  As  a 
matter  of  fact,  we  have  seen  at  the  beginning  of  this  paragraph 
that  the  angle  at  the  origin  is  doubled.  Hence  we  must  supple 
ment  theorem  XI  by  the  following  corollary : 

XII.  The  representation  is  not  conformal  at  the  origin,  since  to 
each  angle  which  has  its  vertex  at  the  origin  in  the  z-plane  there 
corresponds  an  angle  twice  as  large  at  the  origin  in  the  w-plane. 

§  18.    The  Function  w  =  zn,  n  a  Positive  Integer 

After  the  detailed  discussion  of  the  function  zz,  the  investiga 
tion  of  powers  with  arbitrary  integral  exponents  presents  no 
new  difficulties.  Let  such  a  function  be  represented  by 

(i)  a/ =  2". 

*  Equation  (13)  shows  only  that  tan  \f/  =  tan  <p.  —  S.  E.  R. 


88  II.   RATIONAL   FUNCTIONS 

As  in  elementary  algebra  this  is  understood  to  be  the  product  of 
n  factors  each  equal  to  z.  Introduction  of  rectangular  coordi 
nates  furnishes  convenient  formulas  only  for  small  values  of  n. 
We  may  conclude  at  once  from  the  method  of  formation  with 
out  actual  calculation  that : 

I.  The  Junction  w  =  zn  is  by  definition  single-valued  over  the 
entire  plane. 

Retaining  the  notation  of  §  17,  we  obtain  by  repeated  appli 
cation  of  (i),  §  6,  in  polar  coordinates: 

(2)  />  =  >•",    $  =  n$. 

To  each  circle  about  the  origin  of  the  s-plane  (r=  const.) 
there  corresponds  a  circle  about  the  origin  of  the  w-plane 
(p  =  const.).  If  we  allow  the  radius  of  the  former  circle  to 
increase  continuously  from  o  to  oo,  then  the  radius  of  the  latter 
takes  on  all  values,  continuously  increasing  from  o  to  oo.  To 
each  straight  line  <£  =  const,  through  the  origin  of  the  s-plane, 
there  corresponds  a  straight  line  \j/  =  const,  through  the  origin 
of  the  ft'-plane ;  but  the  amplitude  of  the  latter  line  runs  con 
tinuously  through  all  values  from  o  to  2  TT  while  that  of  the 
former  takes  on  only  the  values  from  o  to  2  irjn.  It  therefore 
follows  that : 

II.  The   sector  of  the  z-plane  limited  by  the   rays  <f>  =  o  and 
<f>  =  —  is  mapped  continuously  and  uniquely  upon  the  w-plane  by 
the  function  w  =  zn. 

This  mapping  is  also  reversely  unique ;  but  in  this  case  the 
continuity  is  interrupted  along  the  positive  real  axis  of  the  a/ 
plane  in  that  the  two  sides  of  this  axis  correspond  to  the  two 
lines  which  delimit  the  sector  (Fig.  n). 

If  we  let  <j>  further  increase  from  2  TT/«  to  4  IT///,  from  4  w/n 


§  1 8.   THE   FUNCTION  w  =  z*  89 

to  6  TT/«,  •••,  finally  from  f  —  —  V  to  2  TT,  then  the  correspond 
ing  positive  half  of  the  straight  line  \f/  =  const,  sweeps  over  the 
zt/-plane  the  second,  third,  •••,  nth  time.  The  s-plane  can  then 


\\\\\\\\\\\\\\\\\\\\\\\\ 


z-pfane  w-p/ane 

FIG.  ii 

be  divided  into  sectors,  each  of  which  is  mapped  continuously 
and  uniquely  on  the  whole  ft'-plane.  It  therefore  follows  that : 

III.  The  function  u'  =  z*  fakes  on  each  complex  value  w  at 
exactly  n  points  of  the  z-plane. 

The  values  iu  =  o  and  u<  =  oo  form  the  only  exceptions ; 
each  has  an  exception  at  just  one  point,  viz.  s  =  o  and  z  =  oo 
respectively.  All  the  sectors  of  the  s-plane  have  these  two 
points  in  common. 

There  is  a  simple  relation  connecting  the  different  points  z 
which  give  the  same  value  of  w.  To  exhibit  this  relation,  let 
us  designate  by  e  the  (definite)  complex  number 

(3)  e  =  cos(2  v/n)  +  /  sin(2  TT  ;/). 

which  has  the  property  (cf.  I,  §  6)  that 

(4)  «•«!, 

while  the  lower  powers  c,  c2.  c3,  — ,  e""1  are  all  different  from 
each  other  and  from  i.  It  then  follows  from  the  commutative 
law  of  multiplication  that : 

(5)  («*•«)••=*•, 


QO  II.   RATIONAL   FUNCTIONS 

in  which  k=  i,  2,  •••,  n—  i.     This  result,  on  the  basis  of  defi 
nition  IV,  §  1 7  is  stated  as  follows  : 

IV.  The  function  w  =  zn  is  an  automorphic  function  ;  it  remains 
unchanged  when   subjected  to   the   linear  transformations   of  the 
variable : 

(6)  z'  =  f.k-z      where  k=  i,  2,  •••,  n. 

Since  the  following  theorem,  resulting  directly  from  the  defi 
nition  of  an  automorphic  function,  is  entirely  general,  we  can 
find  relations  connecting  these  n  transformations  : 

V.  When   an   automorphic  function  f(z)    remains    unchanged 
under  two  linear  transformations  of  the  variable,  d  =  ^(z)  and 
z'  =  <£2(X),  it  also  remains  unchanged  under  the  linear  tra?isforma- 
tions  z'  =  <£i  [^C-2)]  and  z'  —  $2  [$1(2)]  compounded  from  them. 

By  means  of  the  definition  of  a  group  of  transformations 
(VII,  §  14),  this  theorem  is  stated  as  follows: 

VI.  The  linear  transformations  of  the  variable  under  which  an 
automorphic  function  remains  unchanged  always  form  a  group. 

The  application  of  this  to  our  example  is  simple  :  If  we  put 
z'  =  e*  •  z  and  z"  =  e'  •  z1,  it  follows  that  z"  =  ck+l  •  z,  which  like 
wise  comes  under  (6)  on  account  of  (4).  Moreover,  we  can 
make  a  still  more  precise  statement  about  the  structure  of  this 
group ;  we  see  that  all  the  linear  transformations  belonging  to 
the  group  can  be  obtained  by  repetition  of  the  first  transforma 
tion.  Hence  the  definition : 

VII.  A  group,  all  of  whose  operations  can  be  formed  by  repeti 
tion  of  a  definite  one  of  them,  is  called  cyclic ;  * 

and  we  have  thus  the  theorem : 

VIII.  The  function  w  =  zn  determines  a  cyclic  group  of  linear 
transforma  tions. 

*  A  cyclic  group  of  transformations  is  a  transformation  with  all  of  its  powers, 
positive  and  negative. —  S.  E.  R. 


§  1 8.    THE   FUNCTION  w  =  zn  91 

Theorem  II  may  also  be  stated  as  follows : 

IX.  The  sector  of  the  z-plane  limited  by  the  rays  <£  =  0  and 
^>  —  2  ir/n  is  a  fundamental  region  for  the  w-plane. 

We  shall  next  take  up  the  question  passed  by  in  the  above 
paragraphs  as  to  how  far  such  a  fundamental  region  is  really 
arbitrary.  Evidently  we  can  take  away  an  arbitrary  section 
from  one  of  its  borders,  providing  we  add  the  corresponding 
section  to  the  other  border.  The  origin  must  always  remain 
on  the  boundary  since  each  transformation  of  the  group  trans 
forms  it  into  itself  and  thus  the  n  fundamental  regions  have  this 
point  in  common  in  whatever  way 
the  first  fundamental  region  is 
chosen.  Moreover,  the  fundamen 
tal  region  must  always  extend  to 
infinity.  But  we  can  bound  it  on 
one  side  by  an  arbitrary  curve  run 
ning  from  the  origin  to  infinity  pro 
viding  this  curve  is  not  intersected 

by  the  curve  obtained  from  the  first  one  by  turning  it  about  the 
origin  through  the  angle  2-jr/n  (cf.  Fig.  12).  Among  all  such 
curves  which  ones  shall  we  now  choose  as  the  best  suited  to 
bound  the  fundamental  region  ? 

There  is  in  fact  no  general  answer  to  this  question  for  all 
automorphic  functions.  But  the  function  w  =  zn  belongs  to  a 
special  class  of  such  functions  for  which  this  question  can  be 
definitely  answered.  It  has  the  property  that  two  conjugate 
complex  values  of  the  function  belong  to  every  pair  of  conjugate 
complex  values  of  the  argument ;  in  particular,  real  values  of  the 
function  belong  to  real  values  of  the  argument.  Thus,  when  we 
take  a  region  in  the  s-plane  which  is  mapped  by  the  function 
w  =  zn  on  that  a^half-plane  with  imaginary  part  positive,  or  "  the 


92  II.    RATIONAL   FUNCTIONS 

positive  w-half -plane,"  then  a  region  symmetrical  to  that  one 
with  reference  to  the  jr-axis  is  mapped  on  "  the  negative  w-half- 
plane."  Hence  we  can  construct  a  fundamental  region  in  the 
following  manner :  locate  first  all  those  lines  to  which  the  parts 
of  the  w-axis  of  reals  correspond  ;  for  the  present  case  they  are 
the  2.n  rays  <j>  =  kir/n  (where  &  =  o,  i,  2,  •••,  2/2—1);  these 
lines  divide  the  z-plane  into  a  certain  number  of  regions.  In 
each  such  region  the  sign  of  the 
imaginary  part  of  w  is  constant. 
For,  on  account  of  the  con 
tinuity,*  it  can  only  change  its 
sign  when  it  passes  through 
zero,  and  this  according  to  hy 
pothesis  is  the  case  only  on  the 
boundary  of  the  region.  More 
over,  every  such  region  for  which, 
for  example,  the  imaginary  part 
of  w  is  positive,  is  mapped  on 

the  whole  positive  half-plane  of  w.  For,  if  it  were  mapped  on 
only  a  part  of  this  half -plane,  its  boundaries,  on  account  of  the 
continuity,  would  have  to  be  mapped  on  the  boundaries  of  this 
part,  which  is  contrary  to  the  hypothesis.  The  s-plane,  then,  is 
divided  into  2  n  half-regions.  In  the  case  at  hand  these  are 
alternately  congruent  and  symmetrical ;  in  more  general  cases 
direct  and  inverted  circle  transformation  is  used  resp.  in  place 
of  congruent  and  symmetrical.  Any  two  of  these  regions  adja 
cent  to  each  other  make  up  a  fundamental  region  answering  all 
of  the  conditions.  Accordingly  : 

X.  An  automorphic  ftinction  which  takes  on  conjugate  values  of 
the  function  at  conjugate  points  is  called  a  symmetric  automorphic 
function. 

*  The  question  of  continuity  is  taken  up  again  in  detail  in  §  31. 


§  1  8.   THE   FUNCTION  w  =  z*  93 

XI.  To  a  symmetric  automorphic  function  corresponds  a  division 
of  the  z-plane  info  alternate  regions  determined  resp.  by  direct  and 
inverted  circle  transformations.      These    regions  are  such  that  any 
two  of  them  adjacent  to  each  other  form  a  fundamental  region  for 
the  function. 

XII.  ///  the  case  of  the  function  w  =  sn,  these  half-regions  are 
sectors  bounded  by  straight  lines  making  angles  of  ir/n  with  each 
other. 

EXAMPLES 

1.  The  function  f(z)  =  (z1  —  z  +  i)3/02  —  zf  is  unaltered  by 
any  of  the  transformations  of  its  variable  given  by  the  six  sub 
stitutions  of  the  group  0,  i/z,  \—z,  i/(i  —  z),  (z—\}/z,  z/(i  —  z). 
It  is  therefore  an  automorphic  function.     This    group  is    also 
finite  discontinuous  (cf.   §  22). 

2.  Show   that    i,    A(z)  =  <o(s),    £(z)  =  u\z)    (where    o>   is    a 
primitive  cube  root  of   unity)  form  a  cyclic  group  of  order  3 
(where  the  order  of  the  group  is  defined  as  the  number  of  trans 
formations  contained  in  the  group). 

3.  Show  that  the  following  transformations  form  a  group  : 


•  •-,  Akn(z)  =  z  -f  nk,  etc., 
where  ;/  =  o,  ±  i,  ±2,  •••,  ±  oo. 

Is  this  group  cyclic  and  what  is  its  order  ? 

4.  A  transformation  is  called  periodic  with  the  period  n  if  the 
identical  transformation  is  obtained  after  applying  the  transfor 
mation  n  (but  not  less  than  ri)  times. 

5.  If  a  linear  transformation  is  of  the  form 

—  £1 

—  £2 

where  &,  £2  are  the  fixed  points  of  the  substitution,  it  is  periodic. 


94  II.   RATIONAL  FUNCTIONS 

If  the  fixed  points  of  the  linear  substitution  coincide,  it  is 
called  parabolic.  If  the  fixed  points  are  distinct,  there  are  three 
classes  of  substitution  as  follows  :  when  the  multiplier 


14, 


is  a  real  positive  quantity,  the  substitution  is  called  hyperbolic. 
When  this  multiplier  has  its  modulus  equal  to  unity  and  its 
amplitude  different  from  zero,  it  is  called  elliptic.  If  the  multi 
plier  has  its  modulus  different  from  unity  and  its  amplitude  not 
zero,  it  is  called  loxodromic.  For  the  substitutions  with  real 
coefficients  only  the  first  three  classes  occur.  These  substi 
tutions  are  often  called  real. 

The  quadratic  equation  (12),  §  14,  which  determines  the  com 
mon  points  of  a  real  substitution  has  real  coefficients  ;  according 
as  the  roots  of  this  quadratic  are  real,  equal,  or  imaginary  the 
real  substitution  is  found  to  be  hyperbolic,  parabolic,  or  elliptic. 
(These  names  are  due  to  KLEIN,  Math.  Ann.,  Vol.  XIV,  p.  122.) 

In  discussing  the  different  cases  we  put  (a  —  d^f  +  4  be  =  M 
from  the  solution  of  (12),  §  14.  Thus 

a~c        a  -\-d— 


and  take  ad—bc=  i  (without  loss  of  generality). 

For  real  elliptic  substitutions,  £j  and  £2  are  conjugate  imagin- 
aries  ;  hence  M=  (a—  d)2  +  4  be  <  o  or 

(a  +  dy  <  ^(ad-bc]  <  4. 
Therefore  k,  using  ad—  be  =.i,  becomes 


k  =  \[_(a  +  d)*-2-  i(a  +  d)  V4  -  (a  +  O2]. 

The  amplitude  of  k  is  thus  cos"1^^  -M)2—  i]  and     k 
denoting  this  angle  by  a  we  now  obtain 


§  1 8.   THE   FUNCTION  w  =  *" 


95 


If  then  wp  be  the  variable  after  /  applications  of  this  substi 
tution,  we  have, 


When  6  is  commensurable  with  2  TT  so  that 


"   —  - 
2  TT      r 


we  have,  taking/  =  r, 


that  is,  WT  =  z ; 

that  is,  the  substitution  is  periodic. 

But  if  0  is  not  commensurable  with  2  TT,  then,  by  a  proper 
choice  of  /,  the  amplitude  pB  can  be  made  to  differ  from  an 
integral  multiple  of  2  ?r  by  a  very  small  quantity  and  leads  to 
an  infinitesimal  substitution. 

6.  It  is  now  evident  that  for  the  elliptic  substitution    a  z- 
circle  through  £t  and  £2  and  its  center  therefore  on  the  ^-axis 
transforms  into  a  w-circle  through  &  and  £2  cutting  the  s-circle 
at  an  angle  a. 

7.  As   an   illustration  of  the 
periodic  character  of  the  elliptic 
transformation  let  us  take  the 
unit  circle  ACBDA  having  its 
center  at  the  origin.    Draw  the 
diameter  AB  along  the  _y-axis. 
Then  the  semi-circle  ACB  can 
be  regarded  as  a  plane  crescent 
of  angle  IT/ 2  and  the  semi-circle 
ABD  as  another  of  the  same 


Q6  II.   RATIONAL   FUNCTIONS 

angle.  Hence  they  can  be  transformed  into  each  other  accord 
ing  to  a  result  due  to  KIRCHHOFF,  Vorlesungen  tiber  mathematische 
Physik,  Vol.  I,  p.  286. 

The  transformation  can  be  most  simply  performed  by  taking 
A(=i)  and  £(=  —  i)  as  the  fixed  points  of  the  substitution, 
which  then  takes  the  form 

w  —  i  _  7     z  —  i 


z-\-i 

The  line  AB  for  the  w-curve  is  transformed  from  the  ^-circular 
arc  ACB;  these  curves  cut  at  an  angle  w/2,  which  is  therefore 
the  amplitude  of  k.  Considerations  of  symmetry  show  that  the 
z-point  C  on  the  .r-axis  can  be  transformed  into  the  ^/-origin 
so  that 


-  i  +  i 
giving  k  =  i  and  the  substitution 

w  —  i  _  .z  —  i 
w  +  i       z  +  i 

It  is  periodic  of  order  4  as  expected  :  it  takes  the  simple  form 

«,=  Z+I  . 

—  z  -\- 1 

Four  applications  of  the  transformation  must  now  give  the 
original  region.  The  first  application  changes  the  interior  of 
ACB  A  into  the  interior  of  ABDA\  a  second  application 
changes  this  latter  region  into  the  region  on  the  positive  side  of 
the  jy-axis  outside  of  the  semi-circle  ADB ;  a  third  application 
transforms  this  latter  region  into  the  region  on  the  negative 
side  of  the  ^v-axis  outside  of  the  semi-circle  ACB ;  a  fourth 
transformation  completes  the  period  and  changes  the  latter 
region  into  the  interior  of  the  semi-circle  ACB  —  the  initial 
region. 


§  i8.   THE   FUNCTION  w  =  zn  97 

8.    Show  that,  if  the  plane  crescent  of  the  preceding  exam 

ple  has    an    angle  of   -  instead  of   -   and   +  /  and  —  /  for    its 
n  2 

angular  points,  then  the  substitution 

z  4-  tan  — 
2  n 


2  n 

is  periodic  of  order  2  n,  and  if  it  be  applied  through  a  period 
to  the  region  of  the  crescent,  will  divide  the  plane  into  2  ;/ 
regions  all  but  two  of  which  must  be  crescent  in  form. 

9.  For  real  parabolic  substitutions  the  quadratic  (12),  §  14 
has  equal  roots  ;  the  fixed  points  of  the  substitution,  £  say,  thus 
necessarily  coincide  on  the  jc-axis.  Thus  M  above  is  zero  and 
(</-|-0)2  =  4  and  d+a  =  2  without  loss  of  generality.  Now 
move  both  origins  to  £,  and  zero  becomes  a  double  root  of  the 
quadratic  so  that  b  =  o  and  a  —  d  =  o.  Hence  a  =  d  =  i  and 
we  obtain 


CZ+l 

1=1+,. 

w      z 

If  the  origins  are  not  moved  to  the  point  £,  then  the  substitu 
tion  is 

w  —  £     s-£ 

Show  that  the  equations  of  transformation  in  real  coordinates 
are 


10.    Show  that  a  z-circle  passing  through  the  origin  is  trans 
formed  by  a  real  parabolic  substitution  with  the  origin  for  its 


98  II.    RATIONAL   FUNCTIONS 

fixed  point  into  a  w-circle  passing  through  the  origin  and  touch 
ing  the  2-circle  ;  and  a  s-circle  touching  the  ^-axis  at  the  origin 
is  transformed  into  itself. 

11.  For  real  hyperbolic  substitutions  the  quadratic  has  real 
and  unequal  roots ;  the  fixed  points  for  the  transformation  are 
thus  two  different  points  on  the  ^-axis  ;  M  is  thus  positive  and 
(a  -f  dj-  >  4  and  we  may  take  (a  +  d)  >  2.  Thus  k  is  real  and 
positive  and  the  substitution  is  hyperbolic. 

Take  the  origin  as  one  of  the  fixed  points  and  g  the  distance 
of  the  other;  o  and  g  are  then  the  roots  of  (12),  §  14  with  the 
conditions  that  (ad—  be]  =  i  and  (a  +  d)  >  2.  Therefore  b  —  o, 
a  —  d=cg,  ad=  i,  k  =  a/d,  and  k  is  greater  or  less  than  i  ac 
cording  as  eg  is  positive  or  negative.  Take  k  >  i  and  we 
obtain 


cz  +  d 
where  a  >  i  >  d,  (a  +  d)  >  2,  and  ad  —  i. 

12.  Show,  therefore,  that  a  z-curve,  drawn  through  either  of 
the  fixed  points  of  a  real  hyperbolic  substitution,  touches  the 
w-curve  into  which  it  is  transformed  by  the  substitution. 

13.  Hence  show  from  Ex.   12  that  any  s-circle  through  the 
two  fixed  points  of  the  hyperbolic  substitution  is  transformed 
into  itself. 

NOTE.  —  The  above  results  and  many  others  are  due  to  POINCARE,  Ada 
Math.,  Vol.  i,  p.  i  and  following. 

14.  Discuss  the  transformation 

w=      g+i. 

§  19.    Rational  Integral  Functions 

We    have    already  defined  at  the  beginning  of   this   chapter 
(II,  §  8)  what   is  in    general  to   be    understood  by  a    rational 


§  19-    RATIONAL   INTEGRAL   FUNCTIONS  99 

integral  function  of  a  complex  variable.  If,  in  the  general  ex 
pression  for  such  a  function  of  a  complex  variable,  we  carry  out 
the  indicated  multiplication  of  sums  or  differences  according  to 
the  distributive  law  ((2),  §  3),  and  finally  collect  into  one  ex 
pression  all  the  terms  which  contain  the  same  power  of  z  multi 
plied  by  a  constant,  the  result  may  be  stated  as  in  analysis  of 
real  numbers  (A.  A.  §  20)  in  the  following  theorem  : 

I.  Every  rational  integral  fu  fiction  of  z  can  be  put  in  the  form  : 
(  i  )  /(*)  =  <7o5"  +  a,zn~l  +  a2zn~2  +  —  +  an_^z  +  an. 

The  integer  n  is  called  the  degree  of  the  function,  provided  aG  =£  o. 

By  using  such  rational  integral  functions  the  following 
theorem  (A.  A.  §  24)  in  the  field  of  real  numbers  may  be  proved 
by  means  of  elementary  operations  : 

II.  An  equation  of  the  nth  degree  has  no  more  than  n  roots  — 
unless  it  is  an  identity,  that  is,  unless  all  of  the  coefficients  are  each 
equal  to  zero.     A  v-fold  root  is  counted  as  v  simpk  roots  in  this 
theorem. 

Since  all  of  the  theorems  used  in  the  proof  of  this  theorem 
are  valid  for  complex  numbers  as  well  as  for  real,  it  follows  that 
Theorem  77  is  also  valid  if  u>e  extend  it  to  include  complex  as  well 
as  real  roots.  But  the  explicit  theorem  that  every  equation  of 
the  nth  degree  in  the  field  of  complex  numbers  has  n  roots  is 
not  proved  in  such  a  simple  manner.  We  shall  obtain  it  later 
(VII,  §  44,  and  VIII,  §  46)  in  other  ways. 

But  it  is  possible  to  assign  limits  between  wrhich  the  zeros  * 
oif(z)  are  included.  Let  J/be  a  number  for  which 


(2) 


^  J/for  ,«  =  i,  2, 


*  Cf.  the  paragraph  following  II,  §  20.  —  S.  E.  R. 


IOO  II.   RATIONAL  FUNCTIONS 

it  then  follows  that 


that  is, 


z   —  i 


For  all  values  of  z  whose  absolute  value  is   >  M+  i  ,  this  last 
fraction  is   <  |  z  |n  —  i  <  [  z  n  ;   it   therefore  follows  that  for  all 
such  values  of  z 
(3)  |/00  -«o*"  |  <    «o*"|. 

In  other  words  it  is  true  that  : 

III.  The  absolute  value  of  the  term  of  highest  degree  in   the 
rational  integral  function  f(z]  is  greater  than  the  absolute  value  of 
the  sum  of  all  the  remaining  terms,  for  all  values  of  z  whose  abso 
lute  value  is  greater  by  at  least  i  than  the  number  M  determined 
by  the  inequalities  (2). 

In  particular,  it  therefore  follows  that  : 

IV.  No  root  of  the  equation  f(z]  =  o  can  lie  outside  of  the  circle 
described  about  the  origin  with  the  radius  M-\-  1. 

If,  on  the  other  hand,  an_vzv  is  the  term  of  lowest  degree  in 
the  rational  integral  function  f(z)  which  has  one  coefficient  dif 
ferent  from  zero,  we  can  put 


in  which 


is  a  rational  integral  function  of  (1/2)  of  degree  (n  —  v).     If  we 
wish  to  apply  Theorem  III  to  this,  we  must  define  a  number  m 


§  i9.   RATIONAL  INTEGRAL   FUNCTIONS  IOI 

by  the  inequality : 

-^-    ^  m  for  /£  =  o,  i ,  2,  •••  «  —  v  —  i . 

When  z  and /are  again  introduced  it  follows  that : 

V.  The  absolute   value   of  the   term    of  lowest  degree   in   the 
rational  integral  function  f(z)  is  greater  than  the  absolute  value  of 
the  sum  of  all  the  remaining  terms,  for  all  values  of  z  different  * 

from  zero  whose  absolute  value  is  less  than  (/+  w)'1. 

Just  as  we  obtained  IV  from  III,  we  have  here  from  V : 

VI.  No  root,  except  possibly  z  =  o,  of  the  equation  f(z)  =  o  can 
lie  inside  of  the  circle  described  about  the  origin  with  a  radius  equal 
to  (i  +  m)~\ 

EXAMPLES 

1.  Find  the  limits  of  the  roots  of 

xt  —  x3  —  ?  x~+  15  A- =  o, 

by  making  use  of  Theorems  III-VI. 

Take  J/=  15  ;  the  greatest  ratio  preceding  and  up  to  this 
one  is  7/15.  Therefore  take  m  =  7/i$.  Hence  there  is  no 
root  outside  of  the  circle  of  radius  M  +  i  =  16,  and  no  root 
inside  the  circle  of  radius  i/(i +7/15)  =  15/22.  This  is 
correct  since  the  roots  are  —3,  2  +  1,  2  —  i,  o. 

2.  Find  the  limits  of  the  roots  as  in  Ex.  i  for  the  equation 

x4  -  3  x?  -  14  x-  +  48  x  -32  =  0. 

3.  Show  that  4  cos2(7r/7)  is  a  root  of  z*  —  $  z'1  +  6  z—  i  =  o 
and  find  the  other  roots.  (Math.  Trip.  1898.) 

*  This  limitation  is  necessary  since,  in  going  from  <J>  to  /  (equation  (4)),  we 
multiply  by  «n. 


IO2  II.    RATIONAL   FUNCTIONS 

§  20.    Rational  Fractional  Functions 
If  all  the  terms  of  a  rational  fractional  function  (I,  §  8)  are 

reduced  to   a   common   denominator,  we  obtain  the   following 

theorem  : 

I.    Every  rational  fractional  function  of  z  can  be  represented  as 

the  quotient  of  two  rational  integral  functions  : 


d\  r(z\  _          = 

h(z)      b,z          ,z  ».         m_^z        m 

II.  The  larger  of  the  two  numbers  m,  n,  or  their  common  value 
when   they  are   equal  to   each  other,  is   called  the  degree  of  the 
rational  function  r(z). 

At  a  point  %  at  which  g(z)  and  h(z]  are  different  from  zero, 
r(z)  has  a  definite  finite  value  different  from  zero.  At  a  point  ^ 
at  which  h(z)  is  different  from  zero  but  g(z)  =  o,  r(z]  is  also  zero  ; 
and  in  this  case  when  zl  is  a  v-fold  zero  *  of  g(z)  we  say  also  that 
z1  is  a  v-fold  zero  of  r(z).  At  a  point  at  which  g(z)  is  not  zero 
and  h(z)  —  o,  r(z)  =  oo  in  the  sense  of  §  12.  We  define  further  : 

III.  A  point  z±  which  is  a  v-fold  zero  of  h(z]  and  not  at  the 
same  time  a  zero  of  g(z]  is  called  a  v-fold  infinity  (a  v-fold  pole]  f 


It  is  sometimes  convenient  to  use  the  following  form  of  ex 
pression  instead  of  II  and  III  : 

IV.     When  r(z)  can  be  put  in  the  form 

(2)  r(2)  =  (z-Zl)"-nW, 

in  which  r^  denotes  a  function  which  is  finite  and  different  from 
zero  for  z  —  z^  then  v  is  called  the  order  of  r(z)  at  the  point  z^ 

*  Or  zero  point  of  the  function,  that  is,  such  a  value  of  the  variable  which  makes 
the  function  vanish.  —  S.  E.  R. 

f  There  are  other  infinities  besides  poles.  Poles  are  the  simplest  infinities. 
Cf.  also  §43.  —  S.E.  R. 


§  20.    RATIONAL   FRACTIONAL   FUNCTIONS  103 

Accordingly,  at  a  pole  the  order  is  negative,  at  a  zero  it  is 
positive  ;  if  the  function  at  a  point  is  finite  and  different  from 
zero,  then  its  order  at  that  point  is  o. 

We  have  finally  to  consider  an  additional  case,  viz.  where 
g(z)  and  h(z)  have  a  common  zero  ;  it  may  occur  after  the  re 
ductions  indicated  in  Theorem  I.  At  such  a  point  the  value  of 
the  rational  function  itself  is  completely  undetermined  (§  12).  But, 
by  rational  operations  which  do  not  necessitate  a  knowledge  of 
the  zeros  of  g  and  //  (A.  A.  §  23),  we  can  find  the  greatest  com 
mon  divisor  k(z)  of  g(z)  and  h(z)  and  thus  put  r(z]  in  the  form 


in  which  gl  and  /^  designate  rational  integral  functions  which 
have  no  common  divisor  and  therefore  (A.  A.  VI,  §  22)  have  no 
common  zero.  If  therefore  we  put 


(4)  . 

the  equation 

(5)  K*)  =  'i« 

is  true  for  all  points  except  the  zeros  of  k(z).     Moreover,  it  is 
now  permissible  to  add  as  a  definition  that  : 

V.  The  function  r(z)  takes  on  the  values  of  r^(z)  even  at  the  zeros 
of  k(z)  which  values  may  be  zero  or  oc. 

With  this  understanding  we  state  the  following  theorem  : 

VI.  The  order  (IV)  of  a  rational  function  at  any  point  is  equal 
to  the  difference  of  the  orders  of  the  numerator  and  denominator  at 
this  point. 

It  follows  further  from  II,  §  19  that  : 

VII.  A  rational  fractional  function  takes  on  no  value  oftener 
than  its  degree  indicates. 


104  II-    RATIONAL   FUNCTIONS 


EXAMPLES 
1.    If  we  divide 

F(z,  w)  =  (az  4-  bw)(cz  +  dw]  +  ew2  +fw  +g 
by  G(z,  w)  =  (az  +  &w\  (\a    >  o,    |  b  \  >  o), 


both  considered  as  functions  of  z,  we  obtain  of  course  as  quo 
tient  and  remainder 

Q(z)  =  (cz  4-  dw)  ,    G,(z)  =  ew*  +fw  +  g. 

If  ^(3,  ?£/)  and  G(zt  w)  are  both  considered  as  functions  of  w, 
what  are  the  quotient  and  remainder  for  this  division  ? 

2.  Perform  the  division  as  in  Ex.  i  for  the  functions 

F(z,  ui)  =  azw  -f-  bw3-  +  cw  +  d, 
G(z,  w)  =  zw  4-  ^. 

3.  When  the  real  axis  is  transformed  into  itself  by  a  linear 
transformation,  it  is  sufficient  that  the  coefficients  of  the  trans 
formation  are  all  real.     Is  this  condition  always  necessary  ? 

§  21.    Behavior  of  Rational  Functions  at  Infinity 

There  are  two  meanings  to  be  attached  to  the  equation 
z'  =f(z).  We  have  usually  regarded  it.  as  establishing  a  rela 
tion  between  two  different  points  of  the  same  or  different 
planes.  But  another  interpretation  was  made  in  (10),  §  10, 
according  to  which  such  an  equation  is  used  to  attach  another 
complex  number  to  the  same  point. 

We  shall  make  particular  use  of  this  latter  idea  to  investigate 
the  behavior  of  any  proposed  function  at  infinity.  We  put  : 

(i)  z'  =  -  and  thus  z  =  -f, 

%  J3 

so   that   the   new  complex    number  z'  =  o  corresponds  to  that 
point  of  the  sphere  to  which  the  complex  number  z  —  oc  intn> 


§  21.    RATIONAL   FUNCTIONS   AT   INFINITY  10$ 

duced  in  §  12  has  been  heretofore  attached.  Thus  when  a 
definite  value  is  attached  to  every  point  of  the  sphere  by  the 
f  unction /(z),  we  can  interpret  these  values  as  a  function  of  z' 
and  as  such  designate  them  by  <f>(z').  If  /(z)  be  rational,  we 
need  only  to  replace  z  by  its  value  as  a  function  of  z'  from  (i). 
We  thus  obtain  a  rational  function  of  z' : 

(2)  /(V)  =  *(*'); 


this  can  be  represented  according  to  I,  §  20  as  the  quotient  of 
two  rational  integral  functions.  The  necessary  multiplication 
of  numerator  and  denominator  by  a  power  of  z'  presupposes  of 
course  z'  =£0.  But  since  f(z)  for  z  =  oo  appears  in  the  undeter 
mined  form  oo/oo,  we  may  write  as  a  definition  (cf.  V,  §  20)  that  : 

I.    The  value  of  the  rational  function  f(z)  for  z  =  oo  shall  be 
understood  to  be  the  value  of  the  function  f(i  j  z*}  =  <f>(z')  for  z1  =  o. 

This  leads  to  the  following  result  : 

If  the  numerator  of  a  rational  function 


is  of  higher  degree  than  the  denominator,  then  will 


and  z'  =  o  is  an  (n  —  ;;/)-fold  pole  of  <f>(z') ;  and  therefore,  ac 
cording  to  the  definition  I,/(oo)  =  ao  and  we  say  that  z=  oo  is 
an  (n  —  ;;z)-fold  pole  ot/(z). 
If  ;;;  =  ;/,  then 

(5)  ^^')  =  f7^4       rf5' 

and/(oo)  =  </>(o)  =  a0/^Q  which  is  finite  and  different  from  zero. 


106  II.     RATIONAL   FUNCTIONS 

If,  finally,  m  >  #,  then 


and/(oo)  =  <£(o)  =  o  ;  and  since  in  this  case  z'  =  o  is  an  (m  —  n)- 
fold  zero  of  <f>(z'),  we  say  also  that  z'  =  oo  is  an  (/#  —  «)-fold 
zero  of  /(z). 

By  extending  the  definition  of  order  of  a  function  (IV,  §  20) 
to  2=  oo,  we  find  in  all  three  cases  that  : 

II.  The  rational  function  (j)  is  of  order  m  —  n  at  z  =  oo  ; 

and  further  (granting  the  existence  of  the  fundamental  theorem 
of  algebra,  §§  44,  46): 

III.  The  sum  of  all  the  orders  of  any  rational  function  is  equal 
to  zero. 

§  21  a.    The  Function  w  =  |  (z  -f-  z'1) 

As  the  first  example  of  a  rational  fractional  function  we  con 
sider  the  function  : 

(!)  .- 


Since  it  is  of  the  second  degree,  it  takes  on  each  value  at  two 
and  only  two  points  of  the  plane.  The  relation  between  any 
pair  of  points  at  which  w  takes  on  the  same  value  can  be  easily 
determined  here  —  just  as  for  any  function  of  the  second  de 
gree  :  if 


it  follows  readily  that  either  z'  =  z  or  z'  =  z~l.  The  function  w 
therefore  remains  unchanged  when  subjected  to  the  linear  transfor 
mation  of  the  variable  : 

(3)  ••-.•/»» 

it  is  an  automorphic  function, 


§21  a.    THE   FUNCTION  w  =  ±(z  +  z'l)  IO/ 

This  transformation    is  reducible    to  a  normal   form   by  the 
methods  of  §  14.     For  this  purpose  it  is  only  necessary  to  put 


Equation  (12),  §  14  thus  reduces  to 
(4)  s'-i=o; 

its  roots  are  ±  i,  the  multiplier  k  takes  the  value  —  i,  and  the 
transformation  (3)  can  be  written  in  the  normal  form 

£i~:fi- 

An  auxiliary  variable  Z  may  therefore    be    introduced  by  the 
following  equations  : 


Therefore 

w  --;(Sf+S?)-H£ 

and  conversely : 

(8)  Z'  =  *^i. 

W  +  I 

Moreover  if  we  put 

/  \  TIT-     w  —  i  i  4-  W 

(9)  W=-    — ,     «/  =  -!——, 

w  +  i  i  —  W 

we  obtain : 

(10)  W=Z*. 

Relation  (i)  between  w  and  z  can  therefore  be  replaced  by  the  three 
simpler  ones  (6),  (10),  (<?),  all  of  which  are  functions  which  we 
have  already  investigated. 

Since  all  of  these  representations  are  in  general  conformal,  it 
follows  further  that : 

The  z-plane  is  mapped  conformally  on  the  w-platie  by  the  relation 
(l),  particular  points  excepted. 


108  II.     RATIONAL   FUNCTIONS 

We  obtain  most  easily  a  conception  of  the  conformal  repre 
sentation  determined  by  the  function  w  by  starting  with  the 
relation  between  the  Z-plane  and  the  £F-plane  denned  by  equa 
tion  (10).  The  mapping  on  the  s-plane  is  then  effected  by 
means  of  equation  (6)  and  on  the  w-plane  by  means  of  equation 
(9).  We  saw  in  §  17  that  the  two  half -planes  of  Z  separated 
by  the  real  axis  may  be  regarded  as  fundamental  regions  of  the 
function  W=  Z2.  Each  of  these  regions  is  mapped  by  this 
function  on  the  entire  ^F-plane.  If  we  divide  the  /^-plane  into 
two  half-planes  by  its  real  axis,  the  positive  half  then  corre 
sponds  to  the  first  and  third  quadrants  of  the  Z-plane  and  the 
negative  half  to  the  second  and  fourth  quadrants. 

Moreover,  equation  (6)  in  connection  with  §  15  shows  that 
real  values  of  Z  correspond  to  real  values  of  z  and  that  pure 
imaginary  values  of  Z  correspond  to  those  values  of  z  whose 
absolute  value  is  equal  to  i  ;  and  from  equation  (9)  it  is  evident 
that  the  JF-axis  of  reals  corresponds  to  the  ze'-axis  of  reals. 
The  corresponding  relation  of  the  four  planes  to  each  other  is 
therefore  shown  in  the  following  figures ;  each  plane  is  divided 
by  the  given  curves  into  a  number  of  regions,  and  those  regions 
which  correspond  to  each  other  are  designated  by  the  same 
letters.  Hence  the  regions  of  the  W-plane  and  of  the  w-plane 
must  each  contain  two  letters,  since  each  of  these  regions  corre 
sponds  to  two  different  regions  of  the  s-plane  and  the  Z-plane. 

To  carry  out  the  representation  more  in  detail,  we  map  other 
lines  of  the  2-plane,  according  to  previous  theorems,  in  turn 
upon  the  Z-plane,  the  /^-plane,  and  finally  upon  the  z£/-plane. 
Thus,  for  example,  to  the  axis  of  pure  imaginaries  in  the  z-plane 
corresponds  the  unit  circle  of  the  Z-plane,  to  this  corresponds 
the  unit  circle  of  the  W^-plane,  and  to  this  the  axis  of  pure 
imaginaries  of  the  w-plane.  Accordingly,  each  of  the  regions 
already  mentioned  are  again  divided  into  two  subregions  which 


§  21  a.    THE   FUNCTION  w  =  \(z  +  c"1)  IOQ 

must  correspond  separately  to  each  other.  In  order  to  deter 
mine  which  regions  correspond  to  each  other,  we  need  only  to 
consider  that  moving  along  a  curve,  in  the  z-plane,  for  example, 
in  a  certain  direction  on  that  curve  corresponds  to  moving 
along  the  corresponding  curve  in  the  7£'-plane  in  a  fixed  direc 
tion  on  that  curve  ;  and  then,  since  the  sense  of  the  angle  re 
mains  unchanged  in  this  representation,  a  region  which  lies  to 
the  left  when  moving  along  a  curve  in  a  certain  direction  must 


Z-p 

\V-p/ane 
lane 

A       C 

B       D 
A                      C 

©A                   c 

w-p/ar?e 
D                    A        C       . 

D 

A                                     c 

B       D"9* 

D 

FIG.  13  a-d 

correspond  to  a  region  which  lies  to  the  left  when  moving  along 
the  corresponding  curve  in  a  definite  direction.  If,  therefore, 
the  correspondence  is  found  for  a  pair  of  subregions,  no  choice 
remains  for  the  remaining  ones,  since  neighboring  regions  must 
have  neighboring  regions  corresponding  to  them.  If  the  cor 
respondence  for  all  the  regions  up  to  the  last  is  determined  in 
this  way,  we  obtain  a  final  check  on  the  problem,  since  the  last 
one  must  again  border  on  a  preceding  one.  In  this  way  we 
obtain  the  accompanying  Figs.  13  e-h. 

To  go  still  further  into  details,  let  us  choose  a  definite  system 
of  curves  of  one  plane  such  that  a  curve  goes  through  each 
point  of  the  plane,  and  find  the  corresponding  system  of  curves 
of  the  other  plane.  We  might  fix  in  mind,  for  instance,  the 
straight  lines  through  the  origin  in  the  z- plane  and,  at  the  same 


no 


II.    RATIONAL   FUNCTIONS 


Z-plane 


FIG.  13  e-h 

time,  the  circles  about  the  origin  perpendicular  to  them.  The 
relations  appear  simplest  by  using  polar  coordinates  in  the  z- 
plane  and  cartesian  coordinates  in  the  ?#-plane.  Accordingly, 


(n)        z  =  r(cos  <f>  -\-i  sin 

and 

(12) 

and  therefore 

(13)  u  =  %(r+r~1) 


—  /  sn 


v  =     r— 


If,  in  these  equations,  <£  is  regarded  as  a  constant  and  r  is 
allowed  to  take  on  all  values  from  o  to  oo,  we  obtain  the  para 
metric  representation  of  that  curve  of  the  w-plane  which  corre 
sponds  to  the  rays  of  amplitude  <£  through  the  origin  of  the 
z-plane.  The  equation  of  this  curve  is  obtained  by  eliminating 
the  variable  parameter  r  by  squaring  and  subtracting  ;  we  find 
in  this  way  : 
(14)  J.  -J-.,. 


§21  a.    THE  FUNCTION  vs  }(*  +  «•*)  HI 

This  is  the  equation  of  an  hyperbola  which  has  the  #-axis  as 
major  axis  and  the  #-axis  as  minor  axis.  Its  foci  are  the  two 
points  -f  i  and  —  i. 

But  if  we  now  regard  r  in  equation  (13)  as  constant  and  let 
<f>  take  on  all  values  from  o  to  2  TT,  we  obtain  the  parametric 
representation  of  that  curve  of  the  ov-plane  which  corresponds 
to  the  circle  of  radius  r  about  the  origin  of  the  s-plane.  By 
eliminating  <£,  we  obtain  the  equation  of  this  curve  in  the  stand 
ard  form  : 
fid  '  2 


This  is  the  equation  of  an  ellipse  which  has  its  center,  foci,  and 
direction  of  axes  in  common  with  the  hyperbola  (14).  Ellipses 
and  hyperbolas  with  the  same  foci  are  called  confocal  (cf.  VIII. 
§  17)  ;  therefore  : 

The  circles  about  the  origin  and  the  straight  lines  through  the 
origin  of  tJie  z-plane  correspond  in  ttie  w-plane  to  confocal  ellipses 
and  hyperbolas  with  foci  at  the  points  +  /  and  —  i. 

The  length  of  the  real  semi-axis  of  the  hyperbola  (14)  is 
equal  to 

(16)  |cos*|. 

As  <£  increases  from  o  to  TT,  \  cos  <£  >  first  decreases  from  i  to  o 
and  then  increases  from  o  to  i  .  Each  of  the  hyperbolas  above 
thus  corresponds  in  the  2-plane  to  two  different  straight  lines 
symmetrical  about  the  j'-axis. 

The  length  of  the  semi-major  axis  of  the  ellipse  (15)  is 
equal  to 

(17)  K'+f-1); 

each  of  these  ellipses  corresponds,  therefore,  to  two  different 
circles  of  the  2-plane  whose  radii  are  reciprocals  of  each  other. 


112  II.    RATIONAL   FUNCTIONS 

For  r  =  i,  v=  o.  But  it  is  not  to  be  inferred  from  this  that 
the  unit  circle  of  the  z-plane  corresponds  to  the  entire  real  axis 
of  the  w-plane ;  for,  it  follows  from  the  first  of  equations  (13) 
that  for  r  =  i  there  can  be  only  such  values  of  u  whose  absolute 
value  is  not  greater  than  i.  The  portion  of  the  axis  between 
the  common  foci  of  these  ellipses  and  hyperbolas  corresponds, 
therefore,  to  the  unit  circle  of  the  s-plane  ;  it  can  be  regarded  as 
a  degenerate  ellipse. 

But  v  =  o  when  <£  =  o  ;  the  corresponding  values  of  u  are 
positive  and  at  least  equal  to  i,  as  is  shown  by  an  examination 
of  the  real  function  u=  i/2(r  +  r~l)  of  the  real  positive  variable 
r.  The  positive  half  of  the  real  axis  of  the  z-plane  corresponds, 
then,  to  that  part  of  the  positive  half  of  the  real  axis  of  the  a/ 
plane  from  the  point  w  =  i  to  oo.  Likewise,  the  negative  half 
of  the  real  s-axis  (<f>  =  TT)  corresponds  to  that  part  of  the  nega 
tive  real  w-axis  which  extends  from  the  point  w  =  —  i  to  oo. 
These  two  parts  of  the  real  w-axis  can  together  be  regarded  as 
a  degenerate  hyperbola. 

For  (f>  =  ±  7T/2,  u  =  o  ;  v  takes  on  all  real  values  from  —  oo 
to  +  oo  (+  oo  to  — co  resp.)  when  r  takes  on  the  real  positive 
values  from  o  to  +00:  the  w-axis  of  imaginaries  corresponds 
to  the  two  ^-half-axes  of  positive  and  negative  imaginaries ;  it 
can  also  be  regarded  as  a  degenerate  hyperbola. 

Since  the  mapping  is  conformal,  it  follows  that  these  ellipses 
and  hyperbolas  always  intersect  in  the  same  angle  as  the  corre 
sponding  circles  and  straight  lines  of  the  z-plane ;  that  is,  in  a 
right  angle.  We  have  therefore  proved  the  geometrical  theorem 
that  an  ellipse  and  an  hyperbola  with  common  foci  intersect  at 
right  angles. 

However,  the  mapping  is  not  conformal  at  the  points  z=  ±  i  to 
which  the  points  w  =  ±  i  resp.  correspond.  An  angle  2  TT  of  the 
«/-plane  corresponds  at  these  points  to  an  angle  TT  of  the  z-plane. 


§22.    AN  AUTOMORPHIC   RATIONAL   FUNCTION          113 

§  22.    A  Somewhat  More  Complicated  Example  of  an  Automorphic 
Rational  Function 

We  have  already  defined  an  automorphic  function  in  IV, 
§  17.  If  the  function  is  to  be  rational  also,  then  the  group  of 
transformations  under  which  the  function  remains  unchanged 
(VI,  §  1  8)  can  have  only  a  finite  number  of  transformations 
(VII.  §  20). 

Definition  : 

I.  A  group  which  is  composed  of  only  a  finite  number  of  trans 
formations  is  called  a  finite  discontinuous  *  group. 

Let  z'  =  \(z)  be  a  transformation  of  such  a  group  ;  then  the 
transformations  : 
(i)  \\z) 


compounded  from  it  also  belong  to  the  group  according  to  V, 
§  1  8.  If  it  is  to  be  finite  and  discontinuous,  then  the  transforma 
tions  (i)  cannot  all  be  different  from  each  other;  by  putting, 
therefore,  A"*(s)  EE  *(,), 

or,  what  is  the  same  thing, 


we  introduce  a  new  variable  Z  by  the  equation 

X\z)  =  Z. 

If  X(z)  is  a  linear  transformation,  then  A*(z)  is  also  a  linear 
transformation  according  to  VI,  §  14;  hence  for  any  value  of  Z 
there  is  a  corresponding  value  of  s,  and  it  follows  that  the  re 
sulting  equation  ,  ~  _  7 

A     \^L,  J    =    4Lr 

*  The  term  "  discontinuous  "  is  necessary  here,  since  we  speak  of  "  finite  con 
tinuous  "  groups  in  which  the  word  "  finite  "  does  not  refer  to  the  number  of 
transformations. 


114  IL     RATIONAL   FUNCTIONS 

is  true  for  all  values  of  Z  ;  in  other  words,  it  follows  that  : 

II.  Every  transformation  of  a  finite  discontinuous  group  of 
linear  transformations  has  the  property  that  it  gives  the  original 
transformation  after  a  finite  number  of  repetitions. 

If,  for  example  (cf.  (8),  §  15), 

AM  -(I-*), 

it  follows  that    X2(»  =  i  -  X(z)  =  i  -  (i  -  z)  =  z, 

and  therefore  n—  2  in  this  case.     But  if  (cf.  (9),  §  15) 


it  follows  that 


and  V(,)  =         ^  =  ^i^  =  2) 

and  therefore  n  =  3. 

Writing  the  resulting  equation  in  one  of  the  forms 


shows  further  that  : 

Ha.  For  each  transformation  z'  =  \(z)  of  a  finite  discontinuous 
group  of  linear  transformations  there  is  another  z"  =  /JL(Z)  having 
the  property  that 

(2)  ,*[X(*)]=*andXM*);i=*, 

or,  in  other  words,  such  that  z  =  /JL(Z')  is  the  solution  of  z'  =  \(z) 
for  z.  We  call  p.  the  transformation  inverse  to  X  and  designate  it 
by  X-1. 

Suppose  now  that 

(3)  AO(*)  =  *,    M*),    A2(s),  -  A^O) 


§22.    AN  AUTOMORPHIC   RATIONAL   FUNCTION         115 

are  the  N  different  linear  transformations  of  a  finite  discontin 
uous  group.  If  Xk(z)  be  any  one  of  them,  then  the  N  values 

(4)  AO[X,W],  A^C*)],  -  A^CA^S)] 

are  all,  on  account  of  the  character  of  the  group,  contained 
among  the  JV  values  (3).  But  otherwise  they  are  all  different 
from  each  other.  For  if,  for  example,  At[A*(s)]  =A»[Aik(s)],  this 
relation  must  remain  true  if  we  substitute  the  value  /n*(z)  in 
place  of  z  in  it,  understanding  /^  to  be  the  transformation  in 
verse  to  Afc.  From  the  equation 


thus  formed,  it  would  then  follow  that 

Xi(*)  =  A^s), 

since  Afc[/*fc(s)]  =  z  according  to  the  definition  of  the  transforma 
tion  inverse  to  a  given  one.  But  that  would  be  a  contradiction 
of  the  hypothesis  that  the  N  transformations  (3)  are  all  differ 
ent  from  each  other.  The  N  values  (4)  are  therefore  all  differ 
ent  from  each  other  ;  and  since  they  are  all  contained  among 
the  N  values  (3),  as  already  shown,  we  can  distinguish  them 
from  these  N  values  only  by  their  arrangement.  Let  us  now 
form  any  rational  symmetric  function  of  the  N  values  (3),  for 
example,  the  sum  2t\i(z)  or  the  product  nzAz(z),  and  apply  to  it 
a  transformation  of  the  group  (3)  ;  that  is,  replace  z  in  it  by 
Xk(z).  It  is  transformed  in  this  way  into  the  corresponding 
function  of  the  N  values  (4).  But  since  these  N  values,  as 
proved  above,  are  different  from  the  N  values  (3)  only  in  their 
arrangement,  and  since  the  function  is  symmetric,  it  follows  that 
it  is  entirely  unchanged  by  this  transformation  ;  and  since  this 
is  equally  true  for  every  transformation  of  the  group,  it  follows 
that  it  is  an  automorphic  function  belonging  to  the  group.  We 
have  therefore  proved  the  theorem  : 


Il6  II.     RATIONAL   FUNCTIONS 

III.  Every  symmetric  function  of  the  N  values  (j)  is  an  auto- 
morphic  function  belonging  to  the  group  (j>),  except  when  it  re 
duces  to  a  constant.  This  exception  might  arise  for  some 
known  symmetric  functions,  but  not  in  general  (since  the  values 
(3)  must  then  be  constant).  Thus  actual  automorphic  rational 
functions  belong  to  every  finite  discontinuous  group  of  linear  trans 
formations. 

Such  particularly  simple  functions  are  obtained  as  follows  : 
Let  z0  be  a  fixed  point  of  one  or  more  (k  say)  of  the  transforma 
tions  (3)  ;  that  is,  let 

(5)  2o  =  MSO)  =  M2o)  =  A2(>o)  •  •  •  =  ViOo)  ; 


it  then  follows  that 

(6)  Xr(z0)  =  A^Oo)]  =  ...  =  Xr[X4_1(*0)]. 

Since  Xr,  \r\^  •••,  XrXA_!  themselves  belong  to  the  transforma 
tions  of  the  group,  these  equations  tell  us  that  the  points  into 
which  ZQ  is  transformed  by  the  transformations  of  the  group  are 
coincident  for  each  k  (from  which  it  also  follows  that  k  must  be 
a  divisor  of  IV).  If  now  <f>(z)  is  a  linear  function  of  z  for  which 
ZQ  is  a  zero,  then  Xr~'(2;0)  is  a  zero  of  <£[Ar(s0)]  ;  and  since  by  (II) 
the  set  of  all  the  transformations  inverse  to  the  transformations 
of  the  group  is  identical  with  this  group  itself,  it  follows  that 
the  zeros  of  A._1 

ri  *[AX*)] 

r=0 

are  coincident  for  each  k,  and  that  the  numerator  of  this  func 
tion  is  the  £th  power  of  an  integral  function  of  degree  (N/k)* 
If  <j>(z)  is  further  determined  so  that  also  its  pole  coincides  with 
a  fixed  point  (different  from  ZQ  and  its  transformed  points)  of 

*  We  se"t  aside  the  case  where  one  of  the  points  Xr(sn)  lies  at  infinity;  in  that 
case  the  degree  would  be  depressed.     Cf.  the  example  following. 


§22.    AN   AUTOMORPHIC    RATIONAL   FUNCTION 

one  of  the  substitutions  (i),  then  the  denominator  of  the  prod 
uct  is  a  power  of  an  integral  function. 

Let  us  now  apply  this  to  the  special  case  of  the  group  of  six 
transformations  which  transforms  one  value  of  the  double  ratio 
of  four  points  into  the  other  five.  The  substitution  z'  =  i/z  has 
ZQ~  —  i  for  a  fixed  point  ;  the  substitution  z'  =  z—  i  has  one  at 
infinity.  A  linear  function  for  which  the  first  is  a  zero  and  the 
latter  a  pole  is  z  -f  i  .  It  is  transformed  by  the  substitutions  of 
the  group  into 

,    v       Z+  I  2  Z—  I  2  Z—  I  2  —  Z 

(7)     —  [—  5          2  —  z;  —  ;  -  ;  -- 

z  z  z—  i  i  —  z 

The  product  of  the  six  values,  viz. 


is  therefore  a  function  of  t/ie  double  ratio  of  four  z  points  which 
remains  unchanged  for  any  permutation  of  the  four  points. 

To  construct  a  fundamental  region  for  this  function,  we  start 
from  the  fact  that  it  is  a  symmetric  automorphic  function.  We 
determine,  as  in  XI,  §  18,  those  curves  along  which  F(z)  is  real. 
The  z-axis  of  reals  is  of  course  one  of  these  ;  but  besides  there 
are  those  curves  along  which  two  and  therefore  every  pair  of 
the  six  factors  are  complex  conjugates.  Now  z  +  i  is  conjugate 
to  2  —  z  along  the  line  x  ==  1/2  ; 
to  (z  -h  i)/z  along  the  unit  circle  ; 

to  (2z—i)/(z—i)  along  the  circle  with  its  center  at  i  and 
radius  i  ;  on  the  contrary,  it  is  conjugate  to  each  of  the  two 
remaining  factors  at  only  certain  points.  But  these  three 
curves  and  the  real  axis  divide  the  z-plane  into  twelve  regions  ; 
it  is  sufficient  to  use  any  adjacent  pair  of  these  regions  on 
which  to  map  the  zt'-plane,  and  since  the  function  w  can  take 


iiS 


II.    RATIONAL   FUNCTIONS 


FIG. 14 


no  value  more  than  six  times,  further  division  lines  are  unnec 
essary  ;  and  thus,  as  shown  in  Fig.  14,  we  have  the  complete 

division  of  the 
z-plane  into  funda 
mental  regions  for 
the  automorphic 
function  w.  It 
takes  on  each  com 
plex  value  once  and 
only  once  in  each 
such  region,  as  will 
be  shown  in  later 
theorems  (§  38  ; 
§46). 

This  figure  appears  particularly  obvious  if  we  transform  it 
stereographically  upon  the  sphere  so  that  the  points  of  inter 
section  of  the  two  circles  fall  on  two  points  of  the  sphere  dia 
metrically  opposite  to  each  other.  If  we  take  these  points  as 
poles  of  a  system  of  spherical  coordinates,  the  two  circles  and 
their  common  chord  transform  into  three  meridians  of  the 
sphere,  and  since  the  angle  of  intersection  is  unchanged  in  this 
transformation  (cf.  §  34),  these  three  meridians  intersect  in 
equal  angles.  Moreover,  the  transform  of  the  axis  of  real  num 
bers  must  cut  these  three  meridians  at  right  angles  ;  we  can  so 
determine  the  constants  at  our  disposal  in  the  function  deter 
mining  this  transformation  that  this  transform  becomes  the 
equator  of  the  sphere.  The  twelve  subregions  thus  become 
alternately  congruent  and  symmetrical. 

To  perform  analytically  the  process  indicated  above,  we  find 
the  substitution  z  —  <£(£)  which  determines  this  transformation 
on  the  sphere,  then  replace  A  by  <£(£)  and  the  new  variable  X' 
by  </>(£')  in  the  equations  (y)-(n)  of  §  15,  and  finally  solve  the 


§22  a.    A   FUNCTION   NOT   LINEAR   AUTOMORPHIC       I  19 

resulting  equations  for  £  '.  Very  simple  formulas  thus  deter 
mine  the  group  ;  the  invariant  function  (8)  also  takes  on  a 
simple  form. 

The  scope  of  this  book  does  not  permit  of  a  more  detailed 
investigation  of  finite  discontinuous  groups  of  linear  sub 
stitutions.* 

§  22  a.    An  Example  of  a  Rational  Integral  Function  which  is  not 
Linear  Automorphic 

As  an  example  of  a  simple  rational  integral  function  which 
is  not  transformed  into  itself  by  any  linear  transformation,  we 
shall  treat  the  following  : 

(i  )  w  =  (2  -  3  z)  =  z(z  -  V3)(z  4-  V-j). 

By  dividing  the  function  and  the  independent  variable  into  real 
and  imaginary'  parts  : 

z  =  x-\-  iy,       w  =  u  4-  Wj 
we  obtain  : 

(2)  u  =  x?  —  3  x\*  —  3  x  =  x(xl  —  3  f  —  3), 

v  =  3  x2y  —  /  —  3  _y  =  ;<3  x*  -  /  —  3)- 

Let  us  now  give  to  z  the  values  on  the  axis  of  real  numbers  ; 
that  is,  put  y  =  o  and  let  x  take  on  all  values  from  —  oo  to  +  °Q. 
For  such  values  w  is  also  real,  since  y=o  gives  v  =  o.  The 
variable  w,  however,  takes  on  some  of  the  values  on  the  real  a/ 
axis  more  than  once,  since  for  y  =  o,  the  following  equation  : 


shows  that  w  is   an   increasing  function  for  z  increasing  only 

*  For  a  detailed  account  of  this  theory,  see  F.  KLEIN,  Vorl.  iiber  das  Ikosaeder, 
Lpz.,  1884. 


120  II.    RATIONAL   FUNCTIONS 

while  —  i  >  z  >  i.  For  z  =  —i,  w  =  +  2,  and  for  z  =  +  i, 
w  =  —  2  ;  therefore  : 

If  z  increases  for  real  values  from  —  oo  through  —  2  to  —  i, 
then  the  real  values  of  w  run,  continuously  increasing,  from  —  oo 
through  —2  to  +  2.  And  if  z  increases  again  from  —  i  to  -\- 1, 
w  remains  real,  but  decreases  to  —  2.  Finally,  if  z  increases  from 
+  /  through  +  2  to  +  oo,  w  increases  from  —2  through  +2  to 
+  06. 

Therefore,  only  one  real  value  of  z  belongs  to  each  real  value 
of  w  whose  absolute  value  is  greater  than  -(-  2  ;  on  the  contrary, 
for  each  real  value  of  w  between  —  2  and  +  2,  there  are  three 
different  real  values  of  z  which  belong  respectively  to  the  three 
intervals  (-2,  -  i),  (-1,  +  i),  (+  i,  +2). 

But  there  are  real  values  of  w  for  other  values  of  z.  For, 
according  to  the  second  of  equations  (2),  z;  is  equal  to  o  if 

(4)  3*2~/-3  =  o; 

and  this  means  geometrically  that  z  lies  on  the  curve  repre 
sented  by  this  equation.  This  curve  is  an  hyperbola  whose 
vertices  are  the  points  x  =  —  i  and  x  =  +  i ,  and  whose  asymp 
totes  cut  the  ;r-axis  at  angles  of  ±  60°.  To  study  the  points 
of  this  hyperbola,  u  may  be  expressed  in  terms  of  x  alone ;  to 
find  this  expression  we  merely  take  the  value  of  y  from  equation 

(4)  and  introduce  it  in  the  first  of  equations  (2),  avoiding  in 
this  wray  the  extraction  of  roots.     We  obtain : 

(5)  u  =  x(^  -  9  x<i  +  9  -  3)  =  -  2  *(4  x1  -  3)- 

Two  points  of  the  hyperbola  with  the  same  abscissa  furnish  the 
same  real  w.  The  equation  also  shows  that  when  z  takes  on 
the  values  on  the  left  branch  of  the  hyperbola  from  infinity  to 
its  intersection  with  the  #-axis,  w  or  u  decreases  from  -f  oo  to 
-\-  2  ;  but  if  z  takes  on  the  values  on  the  right  branch  of  the 


§22  a.    A   FUNCTION   NOT   LINEAR   AUTOMORPHIC       121 


hyperbola  from  infinity  to  the  vertex,  w  increases  from  —  oo  to 
—  2.  Thus  for  any  real  value  of  w  for  which  equation  (i)  has 
only  one  real  root,  there  are  also  two  conjugate  complex  roots. 
But  this  exhausts  all  the  values  of  z  which  furnish  real  values 
of  w.  Hence  equation  (i)  has  either  three  real  or  one  real  and 
two  complex  roots  for  real  values  of  w  excepting  —  2  or  -f  2. 

The  s-plane  is  divided  into  six  regions,  shown  in  Fig.  14^,  by 
the  three  curves  whose  points  furnish  real  values  of  w.  All  the 
points  z  belonging  to  one  of  these 
regions  have  corresponding  values 
of  w  for  which  the  imaginary  part 
iv  has  the  same  sign  ;  or  briefly : 
the  positive  or  the  negative  ze/-half- 
plane  corresponds  to  each  of  these 
regions.*  For,  v  as  a  continuous 
function  of  x  and  y  cannot  pass 
from  positive  to  negative  values 
without  going  through  zero.  But, 
as  we  have  seen,  it  is  zero  only 
when  z  crosses  one  of  the  curves 
which  bound  adjacent  regions.  To 


FIG.  14  a 


determine  whether  a  certain  region  corresponds  to  the  positive 
or  to  the  negative  ^/-half-plane,  we  consider  merely  the  corre 
sponding  directions  in  which  we  move  along  the  curves  that 
bound  this  region  and  the  corresponding  half-plane.  For  ex 
ample,  if  we  move  along  the  boundary  of  the  region  designated 
by  C  from  z  =  —  oo  along  the  z  real  axis  to  z  =  —  i  and  then 
return  to  infinity  along  the  hyperbola,  the  region  C  thus  re- 

*  From  the  preceding  it  has  been  proved  only  that  one  of  the  given  regions  of 
the  2-plane  corresponds  to  a  region  lying  entirely  in  the  positive  or  entirely  in  the 
negative  w-half-plane.  That  this  region  covers  the  corresponding  w-half-plane  com 
pletely  will  be  first  taken  up  in  a  later  theorem  (VIII,  §  38). 


122  II.    RATIONAL   FUNCTIONS 

mains  on  our  left;  the  region  corresponding  to  it  in  the  w-plane 
must  then  also  remain  on  our  left  when  we  move  along  the  cor 
responding  curve.  But  this  corresponding  curve  runs  from 
—  oo  to  +  2  and  then  from  there  to  -f-  oo.  On  the  left  of  this 
path  lies  that  7£/-half-plane  for  which  the  imaginary  part  of  w  is 

positive.     This    therefore    corre- 
w-p/ane 

spends   to  the    region   C  and    is 

accordingly   designated  by  C  in 
ACE 


Fig. 


7?     D    J5* 

We  can  treat  in  the  same  way 

each  of  the  six  regions  into  which 

the  z-plane  is  divided ;  but  this  is  not  necessary,  since  v  changes 
sign  in  crossing  either  the  real  z-axis  or  the  hyperbola ;  and  thus 
any  two  regions  adjacent  to  each  other  in  the  s-plane  correspond 
to  the  two  different  ze/-half-planes.  Therefore,  whenever  the 
region  corresponding  to  C  is  found,  the  w-half-planes  corre 
sponding  to  the  remaining  regions  can  be  determined  success 
ively  ;  we  obtain  a  check  on  the  result  when  at  the  conclusion 
we  shall  have  returned  to  C. 

Further  details  are  obtained  by  dividing  each  of  the  w-half- 
planes  into  two  quadrants  by  the  w-axis  of  pure  imaginaries. 
We  inquire  as  to  what  curves  of  the  s-plane  correspond  to  this 
line  of  division  ;  that  is,  for  those  values  of  z  for  which  w  is 
pure  imaginary,  in  other  words,  for  which  u  =  o.  The  first  of 
equations  (2)  shows  that  this  is  true  for  x  =  o,  that  is,  for  pure 
imaginaries  in  the  z-plane,  and  also  for 

(6)  x*  -  3  /  -  3  =  o, 

that  is,  for  the  points  of  a  second  hyperbola  whose  vertices  are 
the  points  x  —  ±  V3  and  whose  asymptotes  are  inclined  at  angles 
of  ±  30°  to  the  #-axis.  These  curves  divide  each  of  the  six  first- 
mentioned  regions  of  the  z-plane  into  two  subregions,  each  of 


§22 a.    A   FUNCTION   NOT   LINEAR   AUTOMORPHIC       123 

which  corresponds  to  a  quadrant  of  the  w-plane.  To  determine 
the  quadrant  to  which  each  subregion  belongs  we  consider 
merely  the  bounding  curve  and  use  results  already  obtained. 
For  example,  if  the  region  designated  by  A±  borders  upon  a 
part  of  the  positive  real  axis  of  the  s-plane  to  which  the  positive 
real  axis  of  the  a'-plane  corresponds,  then  the  region  A^  can 
only  correspond  to  the  first  quadrant  of  the  &'-plane.  When 


z-  plane 


FIG.  14  c 

this  one  is  determined  we  can  find,  as  before,  the  quadrant  to 
which  each  of  the  remaining  regions  of  the  s-plane  belongs  ;  we 
have  here,  too,  several  checks  on  the  process,  inasmuch  as  regions 
with  which  we  end  border  on  some  already  considered. 

To  find  the  curves  of  the  w-plane  which  correspond  to  other 
curves  of  the  s-plane,  it  is  found  best  to  express  x  and  y  in  the 
equation  of  the  curve  as  functions  of  a  parameter  (eventually 
one  of  the  coordinates  itself  might  be  taken  as  a  parameter). 
If  this  expression  is  then  introduced  in  equations  (2),  we  ob 
tain  a  parametric  representation  of  the  corresponding  curve  in 
the  av-plane. 

Conversely,  to  find  the  curve  of  the  s-plane  corresponding  to 


124  II.     RATIONAL   FUNCTIONS 

a  given  curve  of  the  w-plane,  we  merely  substitute  the  express 
ions  (2)  for  u  and  v  in  the  equation  of  the  first  curve  given  in 
cartesian  coordinates  ;  the  equation  of  the  desired  curve  in  x 

and  y  is  thus  obtained.     But  we 
W~P  must  also  investigate  whether  all 

1  points  on  this  curve  have  corre 

sponding    points    on    the    given 
curve  in  the  w-plane. 


A,  C,  E,. 


A1   Q    Et  But,  very  little  information  con- 

*•      cerning   the    map   of    one    plane 


A   ^  Upon    the   other    is    obtained   by 


\  A 

the  study  of  such   curves.     For, 
apart  from  the   above   examples 

discussed  in  detail,  we  obtain  in 
FIG. 14  d 

the  simplest  cases  curves  whose 

properties  are  not  known  from  elementary  analytical  geometry. 
On  the  contrary,  the  map  determined  by  the  function  can  be 
used  to  facilitate  the  study  of  the  properties  of  such  curves.  It 
gives  direct  information  as  to  how  a  curve  of  one  plane  behaves 
with  respect  to  the  regions  indicated  by  letters  in  our  figures 
just  as  soon  as  we  know  the  curve  of  the  other  plane  corre 
sponding  to  it. 

At  this  point  we  discontinue  the  investigation  of  rational  func 
tions  of  a  complex  variable  and  take  up  the  study  of  the  tran 
scendental  functions.  Just  as  in  the  first  chapter  the  elementary 
operations  on  real  numbers  were  applied  to  complex  quanti 
ties,  we  now  inquire  whether  there  are  not  also  functions  of  a 
complex  variable  which  share  the  fundamental  properties  of  the 
elementary  transcendental  functions  of  a  real  variable.  The  fol 
lowing  chapter  will  serve  as  a  preparation  for  the  answer  to  this 
question. 


§22  a.    A   FUNCTION   NOT   LINEAR   AUTOMORPHIC       125 


MISCELLANEOUS   EXAMPLES 

1.  Determine    the    linear    fractional    transformation     which 
maps  the  points  z  =  z^  z2,  %,  respectively,  into  the  points  z'  =  o, 

I,    00. 

2.  By     means     of    the 
accompanying     figure     in 
which  A  and  A(  ',  etc.,  are 
corresponding  points,  show 
that  angles  are  inverted  in 
the    transformation  by  re 
ciprocal  radii. 

3.  What  are  the  invari 
ant  circles  for  the  transfor 

mation  z'—i/z?     Discuss  this  example  both  analytically  and 
geometrically. 

[Consider  circles  with  their  centers  on  the  j-axis  and  through  the  points  ±  I  ; 
also  circles  with  their  centers  on  the  .r-axis  and  orthogonal  to  the  unit  circle.] 

4.  Discuss    the    effect   on    the    systems   of   straight   lines  x' 
=  const,  /  =  const.,  by  the  transformation 


5.  Show  that  the  system  of  real  numbers  forms  a  group  with 
respect  to  addition. 

6.  If  z~  +  7C2=  i,  show  that  z.  w  are  ends  of  conjugate  radii 
of  an  ellipse  whose  foci  are  ±  i. 

7.  Show  that  two  fixed  points  on  a  circle  subtend  at  any  two 
inverse  points  angles  whose  sum  is  constant. 

8.  Into  what  curves  is  the  unit  circle  z  •  ~z  =  i  (where  z  and 
z  are  conjugates)  transformed  by  the  successive  application  of 
the  substitution  z'  =  (z—  i)/z? 


126  II.    RATIONAL   FUNCTIONS 

9.  Determine  the  general  form  of  the  transformation  that 
transforms  z  •  ~z  =  i  into  itself  (where  z  and  z  are  related  as  in 
Ex.  8). 

10.  Describe  two  kinds  of  maps  of  the  earth's  surface  which 
are  conformal. 

11.  Show  that  the  function  w  =  i/z  has  a  simpler  geometric 
interpretation  on  the  sphere  than  in  the  plane. 

12.  The  equation 

-r'2         r'2 


which  represents  an  ellipse  with  semi-axes  a,  b,  is  satisfied  iden 
tically  by  x'  =  a  cos  <£,  y'  =  b  sin  <£  in  which  <f>  is  the  eccentric 
angle.  Show  that  if  r,  r'  are  focal  radii  of  an  ellipse  having  x, 
y  as  rectangular  coordinates,  a  and  b  semi-major  and  semi-minor 

axes  resp.,  and   <?  =  —  a—^  —   its   eccentricity,  this   ellipse,  de- 
a 

scribed  in  the  positive  sense,  is   represented  by  the   equation 


If  we  put 

a      i  /     .   i\    b      i  /        i\  ,    .    . 

-  =  -(  p  +  -},  -  =  -(  p  --  )  ,  cos  <{>  -f  i  sm  <$>  —  f, 

e      2\       PJ    e      2\       PJ 

p  is  >  i  and  the  last  equation  takes  the  form 

>  ^r-r'f    .  ,    i\ 
h—    —  p/+-    • 
4     V         P*) 


z= 


13.    Find  the  equations  for  the  hyperbola   corresponding  to 
those  for  the  ellipse  in  Ex.   12. 


CHAPTER    III 

DEFINITIONS  AND  THEOREMS   ON  THE  THEORY  OF  REAL 
VARIABLES  AND  THEIR  FUNCTIONS 

IF  the  elements  of  the  theory  of  one  real  variable  and  its 
functions  are  regarded  as  known,  as  in  particular  the  concep 
tion  of  irrational  numbers  and  limits  (A.  A.  chap.  VI)  and  also 
the  idea  of  continuity  (A.  A.  chap.  IX),  we  can  then  apply  this 
theory  in  various  ways  and  show  the  transition  to  functions  of 
two  real  variables. 

§  23.    Sets  of  Points  on  a  Straight  Line  ;  their  Upper  and  Lower 
Bounds  and  their  Limit  Points 

It  frequently  happens  that  a  finite  or  an  infinite  number  of 
the  real  numbers  (points)  *  of  a  finite  interval  f  are  distin 
guished  by  some  property  not  belonging  to  the  others.  We 
then  say :  A  set  of  numbers  {points)  is  defined  on  that  interval. 
Such  a  set  of  points  is  then,  and  only  then,  regarded  as  defined 
when  it  can  be  determined  whether  or  not  any  point  on  the 
interval  belongs  to  the  points  of  the  set ;  it  is  not  necessary 
that  we  should  be  in  possession  of  methods  to  determine  for 
each  point  on  the  interval  whether  or  not  it  belongs  to  the  set.$ 

*  The  numbers  being,  of  course,  simply  a  notation  for  points.  This  notation  is 
complete  in  view  of  the  scheme  by  which  the  system  of  real  numbers  is  set  into 
one-to-one  correspondence  with  the  points  of  a  straight  line.  Cf.  VI,  §3  and  I, 
$4-  —  S.E.R. 

t  We  call  attention  here  to  the  usual  distinction  between  interval  and  segment. 
A  segment  (a,  b},  for  example,  is  understood  to  be  the  set  of  all  numbers  greater 
than  a  and  less  than  b ;  that  is,  exclusive  of  the  end-points  a  and  b  \  and  an  interval 
(a,  V)  is  the  segment  (a,  £)  together  with  a  and  b.  —  S.  E.  R. 

J  The  terms  class,  collection,  aggregate,  assemblage,  etc.,  are  synonyms  of  set. 
—  S.E.R. 

I27 


128  III.    THE  THEORY   OF   REAL   VARIABLES 

I.  The  number  a  is  said  to  be  the  upper  bound  of  a  set  of  num 
bers  {points}  if  the  number  a  has  the  property  that  every  number 
a  —  e  but  no  number  a  +  e  (e  >  0)  is  exceeded  by  a  number  of  the 
set*     For   example,   A/2  f  is  the    upper  bound  of   all  positive 
numbers  whose    square    is  <  2,  and  i    is  the  upper  bound  of 
proper   fractions.     Similarly,  for   the    lower  bound  of   the  set. 
Then  we  have  the  theorem  : 

II.  A  set  of  points  belonging  to  an  interval  always  has  an  upper 
and  a  lower  bound. 

For,  we  can  divide  the  rational  numbers  on  the  interval  into 
two  classes  such  that  every  number  a  of  the  one  class  will  be 
exceeded  by  at  least  one  number  of  the  set  and  every  number 
A  of  the  other  class  will  be  exceeded  by  no  number  of  the  set 
If  there  is  a  smallest  one  in  class  A  or  a  largest  one  in  class  a 
it  is  the  upper  bound,  the  existence  of  which  has  been  affirmed. 
If  neither  of  these  is  true,  then  the  division  %  a  \  A  defines  an 
irrational  number  a  (A.  A.  §  33),  and  this  is  then  the  upper 
bound. 

If,  among  the  numbers  of  the  set,  there  is  a  largest  one  (as 
is  always  the  case  with  a  finite  set),  it  is.  then  the  upper  bound. 
Otherwise  the  upper  bound  does  not  belong  to  the  set. 

For  the  lower  bound,  corresponding  statements  hold. 

We  shall  also  make  use  of  the  following  expression  : 

III.  A  point  a  is  called  a  limit  point  §  of  a  set  of  points  if  points  || 
of  the  set  always  lie  between  a  —  e  and  a  -f-  tfor  every  positive  e. 

*  Of  course,  as  thus  defined  a  is  the  least  upper  bound ;  that  is,  the  least  num 
ber  which  is  an  upper  bound.  —  S.  E.  R. 

t  With  the  understanding  that  \/2  is  a  number.  —  S.  E.  R. 

I  Known  as  the  DEDEKIND  Cut  or  the  DEDEKIND  Partition.  Cf.  PIERPONT, 
The  Theory  of  Functions  of  Real  Variables,  Vol.  I,  p.  82.  —  S.  E.  R. 

§  Synonyms  of  limit  point  are  accumulation  point,  cluster  point,  limiting  point, 
condensation  point.  —  S.  E.  R. 

||  The  plural  is  essential  here. 


§  23.    POINTS   OX   A   STRAIGHT   LINE  1 29 

For  example,  the  limiting  value  of  a  convergent  sequence  of 
numbers  is  a  limit  point  for  the  numbers  belonging  to  the 
sequence.  As  this  example  shows,  a  limit  point  of  a  set  of 
points  may  or  may  not  belong  to  the  set. 

A  set  of  points  is  not  necessarily  arranged  as  a  convergent 
sequence  of  numbers  (A.  A.  §  37);  but  if  it  contains  a  limit 
point  «,  there  are  then  contained  in  the  set  sequences  which 
converge  to  «  and  whose  numbers  all  belong  to  the  set. 

We  now  introduce  the  theorem  of  WEIERSTRASS  : 

IV.  An  infinite  set  of  points  on  a  finite  internal  has  at  least  one 
limit  point  on  this  internal. 

The  proof  of  this  theorem  depends  simply  upon  the  definition 
of  an  irrational  number  by  a  partition  in  the  system  of  rational 
numbers.  We  can  divide  the  rational  numbers  on  the  interval 
into  two  classes  such  that  every  a  of  the  one  class  is  exceeded 
by  an  infinite  number  of  the  set,  every  A  of  the  other  class  by 
only  a  finite  number  or  by  none.  The  lower  end-point  certainly 
belongs  to  the  class  #,  the  upper  end-point  without  doubt  to  the 
class  A,  and  thus  both  classes  really  exist.  There  is  then  a 
number  «,  rational  or  irrational,  such  that  every  number  smaller 
than  it  belongs  to  a,  every  number  larger  than  it  belongs  to  A. 
For  any  positive  number  e,  therefore,  «  —  e  is  exceeded  by  an 
infinitude  of  numbers  of  the  set,  «  -f  e  by  only  a  finite  number, 
and  hence  infinitely  many  numbers  of  the  set  lie  between  a  —  e 
and  a  +  e  ;  in  other  words,  a  is  a  limit  point  of  the  set. 

Of  course,  the  limit  point,  the  existence  of  which  is  proved 
above,  is  not  necessarily  the  only  limit  point  of  the  set ;  it  may 
have  more  than  one,  in  fact  an  infinite  number  of  them ;  and 
each  point  on  the  interval  may  be  a  limit  point  of  the  set.  This 
latter,  for  example,  is  the  case  for  the  set  composed  of  all 
rational  numbers  and  also  for  the  set  made  up  of  all  the  finite 
decimal  fractions  on  the  interval. 


130  III.    THE   THEORY   OF   REAL   VARIABLES 

Moreover,  as  a  consequence  of  the  above  proof  no  limit  point 
of  the  set  can  be  larger  than  a  designated  above.  We  there 
fore  state  the  theorem : 

V.  Among  all  the  limit  points  of  the  set  there  is  always  a  largest 
one  (and  likewise  a  smallest  one] ;  we  call  that  largest  one  the 
upper  limit  (superior  limit  or  Z),  the  smallest  one  the  lower  limit 
(inferior  limit  or  Z)  for  the  numbers  of  the  set. 

The  theorem  that  a  sequence  of  numbers,  which  increase 
continually  but  not  beyond  every  bound,  must  be  convergent 
(A.  A.  §  40)  is  a  special  case  of  the  one  proved  here.  The 
proof  of  the  latter  theorem  —  as  also  Theorem  II  —  shares  with 
that  special  case  the  property  that  it  presents  no  means  to  actu 
ally  specify  the  numbers  whose  existence  is  proved. 

If  the  upper  bound  of  an  infinite  set  does  not  belong  to  the 
set,  it  is  a  limit  point  of  the  set  and  is  then  of  course  the 
superior  limit  Z.  If  however  it  belongs  to  the  set,  it  is  not 
necessarily  a  limit  point,  and  if  it  is  not  a  limit  point,  then  the 
superior  limit  is  different  from  the  upper  bound. 

EXAMPLES 

1.  Recall   carefully   now   the   precise    definitions   of    upper 
(lower)    bound,    limit    point,    superior    (inferior)    limit   Z    (Z). 
Illustrate  each  by  using  the  following  sets  of  numbers  : 

(0   i»  2,  3. 

(2)  i,  1/2,  1/3,  ..-,  i/n. 

(3)  1,0,  1/2,  1/4,  1/8,  .",  i/2"-2. 

(4)  2,  4,  6,  •••,  2  k. 

(5)  All  rational  numbers  less  than  unity. 

(6)  All  rational  numbers  whose  square  is  less  than  2. 

2.  Given  the   set  -P=\  — h  -   >  m  and  n  positive  integers. 
The  limit  points  of  this  set  form  the  infinite  set  o,  i,  1/2,  1/3,  •••, 


§24.    APPLICATIONS;   CONTINUITY  131 

i/m\  determine  which  of  these  limit  points  belong  to  the  orig 
inal  set.  This  new  set,  that  is,  all  the  limit  points  of  />  is  called 
the  derived  set  of  P  and  is  denoted  by  P.  (The  notion  of  the 
derived  set  was  introduced  by  CANTOR,  Math.  Annalen,  Vol.  V 
(1872),  p.  128.) 

3.  Consider  the  set  of  all  positive  proper  fractions,  that  is, 
the  set  P=    -  ;   ,  q  <  r.     What  are  its  limit  points?     Its  upper 

(lower)  bound  ?     Determine  the  derived  set  P. 

4.  Write  a  set  of  points  whose  limit  points  do  not  belong  to 
the  set. 

5.  Has  every  infinite  set  of  points  a  limit  point  ?     An  upper 
(lower)  bound  ? 

§  24.    Applications  of  the  preceding  Theorems  :  Continuity  on  an 

Interval 

A  function  of  a  variable  is  called  continuous  at  a  point  .TO  if  to 
every  assigned  number  e  >  o,  there  exists  another,  8,  such  that 

(i)  \/(x)  —f(x0)  |  <  e  whenever  |  x  —  XG  \  <  8, 

(A.  A.  §  62)  ;  or  otherwise  expressed  (A.  A.  §  61),  if 

(2) 


If  this  condition  is  satisfied  for  all  points  *0  °n  the  interval,  we 
consider  the  question  :  Is  it  possible  for  an  assigned  e  >  o,  to 
determine  a  8  so  that  the  inequality 

(3)  |/(*0  -/(*.)    <« 

*  That  is,  as  x  approaches  xn,  and  denoted  here  by  the  symbol  x  =  x0.  Cf.  also 
VEBLEN  and  LENNES,  Introduction  to  Infinitesimal  Analysis  (Wiley  and  Sons. 
New  York),  (1907),  p.  60.  —  S.E.  R. 


132  III.    THE  THEORY   OF   REAL   VARIABLES 

is  true  for  all  pairs  of  numbers  XQ,  x^  of  the  interval  which 
satisfy  the  inequality 

(4)  t*-.4ro|<8? 

When  attention  was  first  called  to  the  concept  of  uniform  ap 
proach  to  a  limit  of  a  function  (A.  A.  §  66),  it  was  thought  nec 
essary  to  distinguish  between  "  continuity  at  each  point  on  the 
interval  "  and  "  uniform  continuity  on  the  entire  interval."  But 
it  soon  became  evident  that  a  distinction  of  that  kind  is  not 
necessary  here ;  rather,  the  following  theorem  holds : 

I.  When  an  equation  of  the  special  kind  (2)  is  valid  for  all 
points  on  the  interval,  it  necessarily  holds  uniformly  for  the  entire 
interval* 

Assuming  that  it  were  not  the  case,  we  could  then  choose 
any  sequence  of  numbers  converging  to  zero  as 

(5)  81,82,83,  •'•;  KmSn  =  o, 

and,  for  each  number  of  the  sequence,  find  two  points  x^,  xnl 
on  the  interval  such  that 

(6)  !/(*«) -/(*»o)    >e  and  |  *,*-*,„,    <8n. 

Two  possibilities  would  then  arise : 

Either  there  would  be  only  a  finite  number  of  the  points  xno 
which  are  different  from  each  other.  In  this  case  then  at  least 
one  of  these  points  —  call  it  X — is  such  that  the  inequality  (6) 
is  valid  for  infinitely  many  values  of  n.  Since  by  hypothesis 

*  That  is,  Every  function  continuous  on  an  interval  is  uniformly  continuous  on 
that  interval.  This  is  the  so-called  uniform  continuity  theorem  and  is  due  to  E. 
HEINE,  Crelle,  Vol.  74  (1872),  p.  188.  Notice  also  that  this  theorem  does  not 
hold  if"  segment"  is  substituted  for  "  interval,"  as  is  shown  by  the  function  i/x  on 
the  segment  (o,  i),  which  is  continuous  but  not  uniformly  so.  The  function  is  de 
fined  and  continuous  for  every  value  of  x  on  this  segment,  but  not  for  every  value 
of  x  on  the  interval  (o,  i).  — S.  E.  R. 


§24.    APPLICATIONS;   CONTINUITY  133 

the  8,,  converge  to  zero,  we  can,  for  the  given  value  of  e  and  for 
each  8,  so  assign  another  point  xni  that 

(7)  l/(-vni)  —f(X)  \  >  e  and 


But  this  is  contrary  to  the  hypothesis  that  f(x)  is  continuous 
for  each  value  on  the  interval  and  hence  for  X. 

Or  there  would  be  infinitely  many  of  the  points  ,r,i0  which  are 
different  from  each  other.  They  must  then  have  at  least  one 
limit  point  according  to  IV,  §  23.  Let  X  be  such  a  point  and 
then  for  the  given  e  we  can  find  a  point  .r,lo  of  this  kind  and 
with  it  another  point  xni  such  that 

(8)    |/(*J-/(*JI>«,    *i-*J<V*.  I *„-*!<  $/2, 

and  \x,n-X\<8. 

But  on  that  account  the  two  inequalities : 

(9)        I/GO  -/(*)  1  <  c/2  and  |/(*J  -/(*)  |  <  c/2 

cannot  be  true  at  the  same  time,  and  this  means  that  f(x)  for 
#  =  X  is  not  continuous,  contrary  to  the  hypothesis. 

Since  there  is  a  contradiction  in  each  case  Theorem  I  is 
proved. 

A  second  application  of  the  theorem  on  limit  points  is  the 
proof  of  the  following  theorem : 

II.  A  function  f(x)  which  is  continuous  on  an  interval  actually 
assumes  the  value  of  its  upper  (lower}  bound*  at  least  once  on  that 
interval. 

Let  Y  be  this  upper  bound.  Assuming  that  Y  itself  does  not 
belong  to  the  numbers  of  the  set  considered  here  (that  is,  to  the 
values  taken  by  the  function),  then,  by  the  latter  part  of  §  23, 

*  As  defined  in  I ,  §  23.  —  S.  E.  R. 


134  HI-    THE  THEORY  OF   REAL   VARIABLES 

it  must  be  a  limit  point  of  the  set.     We  can  then  assign  an  in 
finite  sequence  of  values  of  the  function 

(10) 

such  that 
(n) 


The  corresponding  values  of  the  arguments  x0,  x^  x2,  —  need 
not  form  a  convergent  sequence.  But  we  can  form  from  them 
an  infinite  sequence  £0,  £1}  £2>  •••  which  converges  to  a  limit  point 
X  of.  the  set  composed  of  these  arguments.  Then  the  functions 


would  also  have  at  least  one  limit  point  ;  but  since  they  are  all 
contained  among  the  numbers  (10)  and  these  have  only  the  one 
limit  point  Y,  it  must  follow  that 

(13)  lim/(4)=K 

•tap 

But  on  account  of  the  assumed  continuity  of  the  f  unction  f(x), 
it  then  follows  that 

(14)  f(*)=Y.  Q.E.D. 

Finally,  the  theorems  of  the  previous  paragraphs  can  be  used 
as  follows  to  free  from  the  assumption  of  monotony  the  theorem 
"  A  continuous  and  monotonic  function  takes  on  each  value 
lying  between  its  initial  and  final  values  "  (A.  A.  II,  §  65).  If 
f(a)  <  o,  f(&)  >  o,  and  if  it  is  to  be  shown  that  the  function 
actually  takes  on  the  intermediate  value  o,  we  reason  as  follows  : 
ampng  the  values  of  the  argument  for  which  f(x)  is  negative, 
there  can  be  no  largest  one  ;  for,  if  f(c)  <  o,  8  can  be  chosen 
so  small  that  also/(V-}-S)  <  o  (cf.  A.  A.  IV,  §  64).  The  upper 
bound  a  of  the  values  x,  for  which  f(x)  <  o,  must  then  be  nec 
essarily  a  limit  point  for  them,  since  it  does  not  itself  belong  to 
these  values  ;  there  are  then,  among  the  numbers  between  a  —  e 


§25.    SETS  OF  POINTS  IN  THE   PLANE  135 

and  a  where  e  is  arbitrarily  small,  always  numbers  for  which 
f(x)  <  o,  while  for  all  larger  numbers /(.r)  >  o.  The  first,  in 
view  of  the  assumed  continuity,  makes  it  impossible  (cf.  A.  A. 
Ill,  §  39)  that/(«)  be  >  o ;  the  second  in  view  of  the  continu 
ity  makes  it  impossible  that/(«)  be  <  o.  Therefore /(a)  must 
=  o.  Q.E.D. 

We  have  accordingly  the  theorem  . 

III.  A  function  f(x)  continuous  on  an  interval  (a,  b)  fakes  on 
every  value  lying  between  /(a)  and  f(b)  at  least  once  for  some  value 
of  x  lying  between  a  and  b, 

even  without  the  limitation  of  monotony. 

We  may  also  mention  here  a  theorem  valid  under  the  results 
of  §  20  (cf.  A.  A.  I,  §  64) : 

IV.  A  rational  function  is  everywhere  continuous  wiiere  it  is 
finite. 

EXAMPLES 

1.  Consider  the  function  y=x2  on  the  segment  (o,  i).     What 
is  the  upper  (lower)  bound,  the  superior  (inferior)  limit  of  y  on 
this  segment  ?     Are  these  points  also  limit  points  for  the  set  of 
values  of  y  ? 

2.  Consider  the  function  r  =  lim — - —  where  o  <  x  <  2. 

,i=x  A'n  +  I 

Here  y  =  x  for  o  <  x  <  i  ; 

y=i/2  for  x  =  i  ;  and  y  —  o  for  i  <  x  <  2.     Answer,  for  this 

function,  the  questions  of  Ex.  i. 

§  25.  Sets  of  Points  in  the  Plane 

In  considering  two  independent  real  variables  (A.  A.  §  19)  the 
most  convenient  geometrical  interpretation  is  to  regard  them  as 
the  rectangular  cartesian  coordinates  of  a  variable  point  in  the 
plane.  Restrictions  on  the  variation  of  the  two  variables  are 


136  III.    THE  THEORY   OF   REAL   VARIABLES 

suitably  imposed  geometrically ;  thus,  for  example,  we  speak  of 
the  point  representing  the  variable  as  situated  on  a  surface  or  on 
a  curve.  And  too,  for  example,  instead  of  saying  :  "  We  consider 
only  values  of  x  and  y  for  which  (x^  +  j'2)  <  i,"  we  say :  "  We 
consider  only  those  points  within  the  circle  of  radius  i  about  the 
origin." 

But  then  it  is  essential  that  we  define  exactly  what  we  mean 
by  the  words  curve,  surface,  in  order  that  there  may  be  no  un 
certainty  about  the  region  of  validity  for  the  theorems ;  as  we 
already  have  the  conception  of  a  point  as  the  representative  of  a 
number-pair,  we  must  necessarily  proceed  from  that  point  of 
view  (and  not,  whatever  else  might  also  be  possible,  from  solid 
to  surface  and  from  this  to  the  curve  and  to  the  point).  We 
therefore  define  at  present  regions  *  and  curves  as  sets  of  points. 

The  theorems  can  be  stated  more  briefly  by  means  of  the  fol 
lowing  terminology :  f 

I.  All  of  the  points  whose  distance  from  a  given  point  A  is  less 
than  a  given  number  S  is  called  a  neighborhood  of  this  point. 

Instead  of  saying  :  "  We  can  so  determine  8  that  all  points  in 
the  neighborhood  of  A  determined  by  8  have  a  given  property," 
we  say  more  briefly:  "All  points  in  the  neighborhood  (or  in  a 
sufficiently  small  neighborhood)  of  A  have  this  property."  Thus, 
for  example,  the  statement :  "  All  points  of  the  neighborhood 
of  the  point  (a,  b]  belong  to  a  given  set  of  points  "  means  the 
same  as :  "  We  can  so  determine  8  that  all  points  (x,  y)  for  which 


(i)  (*-tf)2  +  (7-^)2<8 

belong  to  that  set  of  points." 

*  In  German  "  Flachenstiicke."  —  S.  E.  R. 

f  For  bibliography  and  an  exposition  in  English,  the  reader  is  referred  to  the 
treatise  by  W.  H.  Young  'and  G.  C.  Young,  The  Theory  of  Sets  of  Points,  Cambridge, 
The  University  Press.  — S.  E.  R. 


§25.    SETS  OF   POINTS   IN   THE   PLANE  137 

We  define  also  a  "  rectangular  neighborhood  of  (a,  by  by  the 
two  inequalities  : 

(2)  \x-a\<&,  \y-6\<&. 

It  is  evident  then  geometrically  as  well  as  analytically  that  all 
points  which  satisfy  inequality  (i)  also  satisfy  inequalities  (2); 
and  conversely,  all  points  which  satisfy  (2)  also  satisfy  the  in 
equality  • 

(3)  V(*- 


which  differs  from  (i)  only  in  having  8  V2  in  place  of  8.  Thus, 
whenever  certain  properties  apply  to  all  the  points  of  a  circular 
neighborhood  of  (a,  b),  they  belong  also  to  all  the  points  of  a 
rectangular  neighborhood;  and  conversely.  On  that  account, 
this  difference  is  immaterial  in  many  cases  ;  we  can  use  that  one 
of  the  two  ideas  which  is  the  more  convenient. 

By  means  of  this  idea  of  neighborhood,  we  can  now  apply  the 
concept,  limit  point  of  a  set  of  points,  to  sets  of  points  in  the 
plane  as  follows  : 

II.  A  point  is  called  a  limit  point  of  a  set  of  points  if,  in  -any 
neighborhood  of  it  (arbitrarily  small),  there  are  always  otJier  points.* 

III.  A  point  is  called  an  inner  point  of  a  set  if  a  neighborhood  of 
the  point  belongs  entirely  to  the  set. 

IV.  A  point  is  called  a   boundary  point  of  a  set  if,  in  every 
neighborhood  of  the  point,  there  are  points  of  the  set  and  also  at 
least  one  point  which  does  not  belong  to  the  set.     (It  is  thus  unde 
termined  whether   or  not  the  point  itself   belongs  to  the  set.) 
Every  limit  point  of  the  set,  which  does  not  belong  to  it,  is  a 
boundary  point  of  the  set. 

V.  A  point  of  a  set  of  points,  which  is  not  a  limit  point  of  the  set, 
is  called  an  isolated  point  of  the  set. 

*  The  plural  is  essential  here  as  in  III,  £  23. 


138  III.    THE   THEORY   OF   REAL   VARIABLES 

VI.  A  set  of  points  which  contains  no  isolated  points  (that  is,  a 
set  whose  points  are  all  limit  points)  is  called  dense  in  itself* 

VII.  A  set  of  points  may  have  the  following  property :   Given 
any  two  points  A,  B  of  the  set  and  a  number  e  (arbitrarily  small)  ; 
if  we  can  always  select  a  finite  number  of  other  points  OF  THE  SET 
so  that  each  of  the  distances. 

AA^  AiA2,  .••  An_±An,  AnB 
is  smaller  than  e,  the  set  is  then  said  to  be  connected. 

Examples  of  such  connected  sets  of  points  are  the  lines  and 
surfaces  of  elementary  geometry.  But  the  set  is  also  connected 
if  particular  points  are  excluded  from  all  the  points  of  the  set, 
for  example,  from  all  the  points  inclosed  by  a  circle ;  and  too 
we  obtain  connected  sets  by  considering  only  those  points  of 
such  a  surface  whose  coordinates  are  rational  numbers,  or  only 
those  whose  coordinates  are  finite  decimal  fractions.  To  pass 
therefore  from  the  conception  of  sets  of  points  to  that  of  the 
curve  or  the  surface,  we  must  exclude  such  possibilities.  For 
this  purpose  we  define  as  follows : 

VIII.  A  set  of  points  which  includes  all  of  its  boundary  points  is 
called  closed. 

For  "  closed  and  dense,"  the  one  word  perfect  is  sometimes 
used. 

The  two  last-named  properties  —  that  of  being  connected  and 
closed  —  belong  to  those  sets  of  points  which,  in  elementary 
geometry,  we  call  curves  and  also  to  those  which  we  call  sur 
faces  (for  example,  to  the  set  of  points  on  the  circumference  of 
a  circle,  as  also  to  the  set  of  points  inclosed  by  this,  the  cir 
cumference  being  part  of  the  last  set;  without  the  circumfer 
ence  the  interior  is  not  a  closed  set).  The  difference  is,  that 

*  In  German  "  in  sich  dicht,"  — S,  E.  R. 


§25.    SETS   OF   POINTS   IN  THE  PLANE  139 

the  curve  contains  no  inner  points  in  the  sense  of  definition  III. 
From  this  point  of  view,  we  therefore  give  the  following  most 
general  definitions  of  curves  and  surfaces  : 

IX.  A  connected  and  closed  set  of  points  is  called  a  region  if  it 
contains  inner  points,  an  arc  of  a  curve  if  it  contains  no  inner 
points  (composed  only  of  boundary  points). 

And  too,  there  are  sets  of  points  containing  boundary  points 
whose  separation  from  the  set  leaves  it  open  (that  is,  not  closed) 
and  others  having  boundary  points  which  may  be  separated 
from  it  and  still  leave  it  closed  (as,  for  example,  a  set  consisting 
of  a  circular  surface  with  one  radius  extended).  In  such  cases 
it  is  usual,  when  possible,  to  so  change  the  definition  of  a  set  of 
points  that  such  points  are  excluded. 

On  the  other  hand,  there  are  points  which  are  naturally  inner 
points  but  which  for  special  reasons  we  discuss  not  as  such  but 
as  boundary  points  ;  for  example,  a  circular  surface  "  cut  open  " 
along  a  radius.  This  must  be  considered  separately. 

But  these  previous  definitions  are  much  too  broad  for  our 
purpose :  not  all  sets  of  points  which  come  under  the  one  or 
the  other  of  these  definitions,  have  for  every  curve  and  surface 
those  properties  which  we  have  been  accustomed  all  along  to 
attribute  to  the  curves  and  surfaces  of  elementary  geometry. 
We  must  therefore  add  further  suitable  limitations. 

For  this  purpose  we  start  from  an  entirely  different  point  of 
view.  The  curves  of  elementary  geometry  can  be  determined 
by  a  so-called  parametric  representation  ;  that  is,  if  such  a  curve 
or  an  arc  of  it  is  given,  two  continuous  functions  <£(/),  ^(/)  of  a 
third  variable  /  can  be  chosen  in  many  ways  so  that  all  the 
points  of  this  arc  of  the  curve  and  only  these  are  obtained  when 
we  put 

(4)  * 


140  III.    THE  THEORY   OF   REAL  VARIABLES 

and  allow  the  variable  t  to  take  on  the  values  on  a  given  in 
terval.  And  too,  this  representation  may  always  be  so  arranged 
that  each  simple  point  of  the  curve  is  obtained  only  once. 

X.  We  can  therefore  in  general  regard  any  set  of  points  defined 
by  ftvo  equations  with  these  properties  as  a  curve. 

This  definition  of  a  curve  is,  in  one  sense  narrower,  in 
another,  broader  than  the  one  given  in  IX.  P'or,  while  a  set 
of  points  defined  by  equations  of  this  form  may  have  inner  points 
if  no  further  limitations  are  applied  to  the  functions  </>  and  i//,  yet 
such  a  pair  of  functions  is  not  always  sufficient  to  represent  a 
connected  and  closed  set  of  points  without  inner  points. 

But  a  formulation  at  least  sufficient  for  our  next  purpose  is 
the  following : 

XI.  In  the  following,  only  those  sets  of  points  which  satisfy  at 
the  same  time  both  definitions  IX  and  X  are  designated  as  curves. 

XII.  In  particular,  we  designate   as  a  simple  curve  that  one 
which  has  no  double  points,  that  is,  one  on  which  there  are  always 
distinct  points  corresponding  to  different  values  of  the  parameter  in 
equation  (4). 

Analogous  to  this  we  stipulate  further  : 

XIII.  In  what  follows  we  designate  as  surfaces  only  those  sets 
of  points  which  satisfy  definition  IX,  and  whose  boundary  points  form 
one  or  a  finite  number  of  simple  curves  (XI]  not  intersecting  in  pairs. 

Further  limitations,  while  not  essential,  are  at  all  events  use 
ful  for  most  of  the  theorems  deduced  later.  We  therefore  define 
further : 

XIV.  If  the  functions  <£(/),  ^(/)  are  continuous  and partitively 
monotonic*  the  curve  is  called  a  path  ;  and  a  surface  bounded  by  a 
path  is  called  a  domain. 

*  In  German  "  abteilungsweise  monoton.  "  In  this  connection  cf.  VEBLEN  and 
LENNES,  I.e.,  p.  50.  — S.  E.  R. 


§  25.    SETS   OF   POINTS   IX   THE  PLANE  141 

In  the  discussion  of  later  theorems  we-  will  be  limited  mostly 
to  paths  and  domains.  To  be  sure,  we  thus  exclude  a  number 
of  cases  which  are  of  interest  in  the  theory  of  functions.  In 
many  cases  it  is  possible  to  discuss  such  curves  and  surfaces  by 
regarding  them  as  the  limiting  cases'of  paths  and  domains,  resp. 
But  the  mere  assumption  of  the  limiting  process  is  not  usually 
sufficient  ;  on  the  contrary,  it  is  necessary  in  drawing  conclu 
sions  to  pass  uniformly  to  the  limit  (A.  A.  §  66).  Hence  the 
following  definition  : 

XV.  If  the  functions  <£„(/),  ^B(/)  satisfy  the  conditions  of  XIV 
for  every  value  of  n,  and  further,  if 

(5)  lim  $„(/)  =  <£(/),  lim  «0 


UNIFORMLY  for  all  values  of  t  under  consideration  inclusive  of  the 
end-values,  then  the  curve  represented  by  equations  (4)  is  called  an 
improper  path,  and  a  surface  bounded  by  a  finite  number  of  such 
curves  is  called  an  improper  domain. 

XVI.  The  theorem  on  limit  points  {IV,  §  23)  is  valid  also  for 
sets  of  points  in  the  plane.  For,  if  we  disregard  the  second  coor 
dinate  of  the  points  of  the  set,  the  results  are  as  in  §  23  ;  that  is, 
a  number  a  can  always  be  found  such  that  infinitely  many  points 
of  the  set  have  a  first  coordinate  lying  between  a  —  e  and  a  +  e 
for  e  arbitrarily  small.  Let  us  now  keep  in  mind  only  these 
points,  and  consider  their  second  coordinate  :  there  is  then  at 
least  one  number  (3  such  that  infinitely  many  of  the  points  just 
determined  have  a  second  coordinate  lying  between  ft  —  c  and 
,8  -h  e.  Together  these  two  statements  tell  us  that  infinitely  many 
points  lie  in  every  neighborhood  of  the  point  («,  /?).  Q.E.D. 

The  conclusion  in  this  form  assumes  that  not  only  the  number 
of  points  themselves  but  also  the  number  of  different  values  of 
their  first  or  their  second  coordinate  is  infinite.  But  this  is  always 


142  III.    THE   THEORY   OF   REAL  VARIABLES 

the  case  excepting  only  when  infinitely  many  of  the  points  have 
the  same  first  or  the  same  second  coordinate ;  but  in  this  excep 
tional  case  they  must  lie  on  a  straight  line  and  then  the  existence 
of  a  limit  point  follows  at  once  from  §  23. 

XVII.  A  domain  is  called  simply  connected  when  any  closed  curve 
in  it  can  be  contracted  to  a  point  by  continuous  deformation  without, 
in  so  doing,  going  outside  of  the  domain*  For  example,  the  sur 
face  of  a  circle  or  of  a  square  is  simply  connected;  but  not  the 
surface  between  two  concentric  circles,  since  a  circle  on  this  sur 
face  concentric  to  the  two  bounding  circles  cannot  be  contracted 
to  a  point  without  going  outside  of  the  surface. 

EXAMPLES 

1.  Is  the  surface  of  a  sphere,  considered  as  the  stereographic 
projection  of  the  points  of  the  plane,  simply  connected?  Do  two 
non-intersecting  spheres,  not  bound  or  joined  together  in  any 
way,  make  up  a  connected  surface  ? 

2.  Let  us  consider  the  area  in 
closed  between  and  completely 
bounded  by  two  concentric  circles. 
It  is  connected  but  not  simply.  We 
can  make  it  simply  connected  by 
setting  an  impassable  barrier.  The 
most  effective  way  to  do  this  is  to 
suppose  the  surface  actually  cut 
along  the  line  of  the  barrier  as  AB  in  the  adjoining  figure. 
The  surface  is  now  a  simply  connected  one. 

3.  Again,  the  surface  of  an  anchor  ring,  not  simply  con 
nected,  can  be  made  so  by  two  barriers.  As  actual  cuts  they 

*For  a  more  complete  treatment  of  connectivity  see  OSGOOD,  Lehrbuch  der 
Funktionentheorie,  Vol.  I,  p.  144,  and  FORSYTH,  Theory  of  Functions,  p.  313. — 
S.  E.  R. 


§  26.    FUNCTIONS   OF  TWO   REAL  VARIABLES 


143 


would  appear  as  in  the  accompanying 
figure. 

[This  method  of  resolving  surfaces  into 
simply  connected  ones  by  the  establishment 
of  barriers  is  that  adopted  by  RIEMANN, 
Gesammelte  Werke,  pp.  9-12  and  84-89.] 

4.  Consider     the     set     of     points 
/>=[oi],   that  is,  all  the  points  on 
the    interval    (o,   i).      What    are    its 

limit  points,  upper  (lower)  bounds  ?    Is  it  dense,  closed  ?    Is  the 
set  of  Ex.  3  at  the  end  of  §  23  dense,  closed  ? 

5.  Are  the  following  sets  dense  in  itself,  closed,  perfect  ? 
(a)    A  segment  not  including  its  end-points. 

(&)    A  segment  with  its  end-points. 
(f)    The  set  of  rational  numbers. 

§  26.    Continuity  of  Functions  of  two  Real  Variables 
I.    (Definition.)     An  equation  of tJie  form 
(i)  lim  lim/(.v,  v)  =c 

signifies  t)ie  same  as      lim  Jlim/(jt:,  y)  \  =  c, 

in  other  words,  the  inner  limit  is  to  be  evaluated  first. 

The  order  of  evaluating  two  successive  limits  of  a  function 
of  t\vo  real  variables  is  not  interchangeable  even  in  simple 
cases ;  for  example,  since 


lim 


but 
(3) 


x—  y  —  x*  +  )*      i—  x 


the 


x  —  y  —  x2-  -f-/ 


^_—    _      T      * 


*It  is  interesting  to  note  that  (x+y)/(x-y)  would  be  sufficient  here,  viz. 


lim 


—  y 


-  and  lim  -  =  i  ;  but  lim  lim 
x  x=0.r  y 


—  y 


—  i.  —  S.  E.  R. 


144  HI.    THE  THEORY   OF   REAL  VARIABLES 

II.    (Definition.)      The  equation 

(4)  lim  f(x>y)=c 

x=a  j/=6 

means  that  for  every  assigned  number  e  >  o,  there  exists  another, 
8,  such  that 

(5)  \f(*,y)-c    <* 

for  EVERY  pair  of  numbers  x,  y  which  are  different  from  a,  b  and 
which  satisfy  the  inequality 


(6)  VO-tf)2  +  (7-^)2<S. 

According  to  the  terminology  of  §  25,  this  definition  is  stated 
as  follows  :  equation  (4)  signifies  that  f(x,  y)  is  infinitesimally 
different  from  c  in  the  neighborhood  of  (a,  1)}  —  the  point  itself 
excepted. 

If  equation  (4)  holds,  equation   (i)  also  holds,  and  too  the 
equation 
(7) 


for  every  A.     But  the  converse  is  not  true  ;  for  example,  while 


(8)  lira  lim  - 

x^    y^ 

the  following 


f  \  r          -  i-X2 


which  is  a  function  of  X.     This  would  not  be  the  case  if  we  had 
here  an  equation  like  (4). 

III.    (Definition.)     If  the  equation 
(10)  lim  f(x,y}=f(a,b} 

x=a  y=b 

holds  for  a  function  of  two  variables,  then  f(x,  y]  is  a  continuous 
function  of  x  and  y  at  the  point  (a,  b). 


§  26.    FUNCTIONS   OF  TWO   REAL   VARIABLES  145 

As  the  example  above  shows,  a  function  of  x  and  y  may  be  a 
continuous  function  of  x  and  also  a  continuous  function  of  y  for 
every  value  of  x  and  y,  and  yet  not  necessarily  be  a  continuous 
function  of  x  and  y  in  the  sense  of  definition  III. 

On  the  contrary,  the  following  theorem  holds  as  for  functions 
of  a  single  variable  (§  24) : 

IV.  If  a  function  of  tu<o  variables  is  a  continuous  function  of 
tJiese  two  variables  at  every  point  of  a  finite  domain,  it  is  also 
(uniformly}  continuous  in  tJie  entire  domain;  that  is,  for  every 
assigned  e  >  o,  there  exists  another,  8,  such  that, 

(11)  LA**  *)-/(*.,  *)!<« 

for  every  pair  of  points  (x^  }\),  (x2,  y2)  of  the  domain  which 
satisfies  the  inequality 


(12)  Vte-arO'+O', -;•!)*<  8. 

From  this  it  follows  further  that : 

V.    If  x,  y  are  continuous  functions  of  //,  v,  and  if  z  is  a  contin- 
iwus  function  of  x,  y,  then  z  is  a  continuous  function  of  u,  v. 
If 

(13)  ?/  =  <£(>,  y),  v  =  ^(x,y) 

are  defined  as  (single-valued)  functions  of  x  and  y  in  a  domain  B 
of  the  jvy-plane,  we  can  interpret  //,  v  as  coordinates  of  points  of 
another  plane.  Each  point  (x,  y]  of  B  will  then  have  a  definite 
point  of  the  7/#-plane  corresponding  to  it  by  equation  (13)  ;  the 
set  of  all  the  points  which  correspond  in  this  manner  to  the 
points  of  B,  determine  a  set  of  points  in  the  //r-plane.  But 
whether  this  set  of  points  also  determines  a  region  is  known  only 
when  more  details  concerning  the  functions  <£,  \f/  are  given.  It 
is  sufficient  here  to  investigate  cases  where  <£,  ^  are  not  merely 


146  III.    THE  THEORY   OF   REAL  VARIABLES 

continuous  functions  of  the  two  variables  x  and  ^but  have  other 
/imitations  given  in  the  course  of  the  investigation. 

We  proceed  indirectly  from  (#,  y]  to  (u,  v)  by  introducing  an 
auxiliary  plane  (£,  rf)  whose  points  have  for  coordinates  one  of 
the  old  and  one  of  the  new  variables  ;  thus : 

(14)  £  =  *",  rj=v  =  ^(x,y). 

We  now  give  to  x  a  definite  value  a  (found  in  J3) ;  geometrically, 
this  amounts  to  considering  a  line  parallel  to  the  jy-axis.  If  this 
parallel  has  only  one  closed  and  connected  segment  (y0,  y^ 
(VII,  §  25)  in  common  with  the  domain  B,  then  rj  is  defined  on 
the  corresponding  interval  as  a  continuous  function  of  y  by 
equation  (14) ;  for,  if  i//  is  a  continuous  function  of  both  vari 
ables,  it  is  a  continuous  function  of  each  separately.  Moreover, 
if  i//  as  a  function  of  y  is  monotonic  on  this  interval,  then  to 
the  interval  (jv0,  yi)  there  corresponds  an  interval  ($(a,  jv0)> 
$(a>  J^i))  sucn  tnat  on  it>  conversely,  y  can  'be  regarded  as  a  con 
tinuous  and  monotonic  function  of  77.  The  interval  we  are  con 
sidering  on  the  line  parallel  to  the  jv-axis  then  has  a  reversibly 
unique  correspondence  with  a  definite  interval  on  a  line  parallel 
to  the  ?7-axis,  that  is,  such  that  not  merely  one  and  only  one 
point  (£,  rj)  corresponds  to  each  point  (x,  y)  but,  conversely, 
one  and  only  one  point  (x,  y)  corresponds  to  each  point  (£,  rf}. 

But  if  the  straight  line  has  two  different  intervals  (y0,  y^  and 
(y2,  jv3)  in  common  with  B,  and  if,  for  example,  i/>  is  monotonic 
increasing  on  each  of  these  intervals,  it  does  not  follow  from  this 
alone  that  \[/(a,  yz)  must  be  >  \J/(a,  j^).  For,  each  of  these  in 
tervals  has  an  interval  on  a  line  parallel  to  the  7^-axis  correspond 
ing  to  it  in  a  reversibly  unique  manner ;  but  these  two  latter 
intervals  may  overlap  so  that  a  part  of  the  interval  thus  deter 
mined  is  "  doubly  covered. "  There  are  therefore  two  points  of 
the  ^-plane  corresponding  to  each  point  of  this  last  part. 


§  26.    FUNCTIONS   OF  TWO   REAL   VARIABLES  147 

But  under  the  first  supposition  let  us  now  consider  a  neighbor 
ing  straight  line  x  =  a  +  h.  If  this  too  has  only  one  interval  in 
common  with  B,  there  is  then  an  interval  on  the  straight  line 
£  =  a  4-  h  corresponding  to  it.  The  end-points  j0,  \\  of  this 
interval  take  on  values  for  x  =  a  +  h  other  than  those  of  the 
corresponding  interval  for  x  =  a.  But  since  B  is  by  supposition 
a  domain,  y0(a  +  h)  and  }\(a  +  //)  differ  infinitesimally  from 
y0  (a)  and  }\(a)  respectively,  for  h  sufficiently  small  ;  and,  on  ac 
count  of  the  prescribed  continuity,  $\_(a  +  h),  Jo(#  +  ^)]  and 
»/>[(#  +  ^),  }\(a  H-  A)]  differ  infinitesimally  from  \f/[a,  y0(a)~\  and 
\l/[a,  }'i(a)]  respectively. 

We  suppose  that  these  hypotheses  hold  for  all  values  of  x 
under  consideration.  Then  two  continuous  functions  of  £,  and 
thus  two  curves  in  the  ^-plane  are  defined,  according  to  the  last 
proof,  by  the  equations  : 


These  curves  have  no  point  in  common,  when  we  suppose  i/f, 
as  above,  to  be  a  monotonic  function  of  its  second  argument, 
since  for  every 

< 


The  set  of  all  the  points  (£,  rj)  for  which 

(1  6)  %(£)  g  r,  g  „(*) 

forms  in  the  ^-plane  a  region  C  which  has  a  reversibly  unique 
correspondence  with  the  domain  B.  Moreover,  the  function 

(I7)  y=e&-n)  =  B(X,v) 

obtained  by  reverting  the  second  equation  in  (14)  is,  for  all  £rj  of 
this  region,  a  continuous  function  of  its  two  variables  and,  for  x 
fixed,  is  a  monotonic  function  of  v.  (The  continuity  with  refer 
ence  to  the  two  variables  is  deduced  from  the  corresponding 


148  III.    THE  THEORY   OF   REAL  VARIABLES 

property   of   \]s,    as    for  functions   of   one    variable    (A.  A.  Ill, 

§  65))- 

If  we  pass  now  from  the  ^-plane  to  the  zw-plane  by  means 
of  the  equations  : 

(18)  u  =  4>(x,  y)  =  <£[£,  0(S,  ,?)]  =/(£  77),   v~i, 

we  can  draw  corresponding  conclusions  if  the  functions  satisfy 
corresponding  hypotheses.  In  doing  so  it  is  only  necessary  to 
notice  the  following  conditions  :  When  any  parallel  to  the  ^-axis 
has  only  one  connected  interval  in  common  with  the  domain 
B,  corresponding  conclusions  for  the  ^plane  cannot  be  drawn, 
since  there  may  be  in  common  with  the  region  C  several  dis 
tinct  intervals  on  a  line  parallel  to  the  £-axis.  Parts  of  the  uv- 
plane  could  then  be  multiply  covered  by  the  points  denned  by 
(13).  This  possibility  must  be  excluded,  and  we  have  then  the 
following  formulation  of  the  results  : 

VI.  If  the  functions  (ij)  are  continuous  in  the  domain  B  and 
such  that  to  tivo  different  points  (x,  y]  of  tJiis  domain  there  are 
always  two  different  pairs  of  values  (u,  v]  ;  if,  further,  if/,  for  a 
given  x,  is  a  monotonic  function  of  y  and*  if  the  function  f  defined  by 
(18)  is,  for  a  given  77,  a  monotonic  function  of  £  :  then  the  points  of 
the  uv-plane  corresponding  to  the  points  of  B  by  (ij)  cover  a  region 
C  of  this  plane  uniquely  without  gaps  ;  and,  conversely,  in  this 
region  x,  y  are  also  continuous  functions  of  u,  v. 

We  thus  say  :  The  domain  B  is  mapped  continuously  on  the 
region  C  by  means  of  the  functions  (/j). 

§  27.    Derivatives 

I.  The  derivative  of  a  function  f(x)  at  a  given  point  x  is  defined 
/v  the  equation  : 


dx      ^  A 


§  27.    DERIVATIVES  149 

provided,  of  course,  that  this  limit  exists.  If  it  exists  for  every 
value  of  -r,  at  least  on  a  given  interval,  then  its  values  on  this 
interval  form  a  definite  function  of  x,f'(x),  which  is  called  the 
derived  function  or  the  derivative  otf(x). 

If  /(x)  is  a  rational  function  of  x,  then  the  function  on  the 
right  side  of  equation  (i)  is  a  rational  function  of  both  the 
variables  x  and  h.  Then,  according  to  IV,  §  24,  for  a  given 
value  of  x,  only  two  cases  can  arise,  viz. :  either  the  function 
increases  beyond  all  bounds  as  h  approaches  zero,  or  the  limit 
exists  ;  but  the  first  case  as  shown  in  elementary  differential 
calculus  occurs  only  when  the  given  value  of  x  makes  the 
denominator  of  f(x)  zero.  Hence  the  theorem  : 

II.  A  rational  function  of  a  real  variable  always  has  a  definite 
derivative  wherever  the  function  is  finite. 

It  is  not  always  necessary  to  apply  the  definition  I  directly 
to  the  function  in  order  to  find  its  derivative,  since,  as  in  the 
differential  calculus,  the  differentiation  of  more  complicated 
functions  can  be  made  to  depend  upon  the  differentiation  of 
simpler  ones.  Methods  for  this  purpose  and  the  derivatives  of 
the  simplest  functions  are  supposed  to  be  known  here. 

We  suppose  it  known  too  that  a  function  of  a  real  variable 
represented  by  a  power  series  has  a  definite  derivative  at  each 
inner  point  on  its  interval  of  convergence  and  that  this  deriva 
tive  can  be  found  by  differentiation  of  the  given  series  term  by 
term  (A.  A.  §  81). 

Finally,  we  also  suppose  it  to  be  known  that  the  deriva 
tive,  provided  it  exists  at  an  inner  point  on  the  interval,  can 
not  be  negative  (positive),  if  the  function  at  that  point  is 
increasing  (decreasing)  for  x  increasing,  and  that  it  must  be 
equal  to  zero  if  the  function  has  at  that  point  a  maximum  or  a 
minimum. 


150  III.    THE  THEORY   OF   REAL  VARIABLES 

III.    The  partial  derivative  of  a  function  f(x,  y)  with  respect  to 
x,  for  y  constant,  is  defined  by  the  eqiiation  : 


dx 

Two  things  are  necessary  for  its  complete  determination,  viz.  ; 
the  determination  of  the  variable  with  respect  to  which  it  is  to 
be  differentiated  and  the  variables  which  are  for  the  process 
regarded  as  constant. 

Rules  for  transforming  such  partial  derivatives  when  passing 
to  new  variables  are  easily  established  arithmetically,  provided 
we  grant  the  existence  and  the  continuity  of  the  partial  deriva 
tives  which  occur  in  the  process.  Under  these  conditions  we 
suppose  such  rules  to  be  known. 

The  hypotheses  of  Theorem  VI,  §  26,  in  which  the  occurrence 
of  the  unknown  function  y  is  somewhat  troublesome,  can  be  re 
placed  by  simpler  but  less  general  ones.  For,  according  to 
those  rules,  we  have, 


•3u\ 


Tj=const. 


and        IT-:.  .  .  .  .        , 

$y )  z=c°nst- 

and  therefore 

dv\  fdu\  _  du     dv  _  dv     du 

dyjx=const.     \d£jr)=const      dx     dy      dx     dy' 

providing  y  is  regarded  as  constant  on  differentiating  with  re 
spect  to  x,  and  x  constant  on  differentiating  with  respect  to  y  on 
the  right-hand  side  of  the  equations.  But  since  continuous 
functions  can  change  sign  only  in  passing  through  zero,  it  fol 
lows  that,  if  the  "  functional  determinant "  on  the  right-hand 


§  28.    INTEGRATION  !$! 

side  of  (3)  is  different  from  zero  in  the  entire  domain  B,  then 
\f/  for  x  constant  is  a  monotonic  function  of  y,  and  f  for  77  con 
stant  is  a  monotonic  function  of  £.  From  VI,  §  26,  it  therefore 
follows  : 

IV.  If  //,  v  in  the  domain  B  are  continuous  functions  of  x  and 
y  with  continuous  first  partial  derivatives  ajid  if  the  functional 
determinant  (j)  is  different  from  zero  everywhere  in  B,  tlien  this 
domain  is  mapped  continuously  by  u,v  on  a  region  C  of  the  uv-plane  ; 
in  fact,  this  region  of  the  //z'-plane  is  thus  covered  everywhere 
uniquely,  providing  that  different  points  of  B  always  corre 
spond  to  different  pairs  of  values  //,  v. 

Conversely,  x,  y  inside  of  C  are  therefore  continuous  func 
tions  of  //,  v  with  continuous  first  partial  derivatives  which  are 
found  by  known  rules. 

§  28.    Integration 

We  must  go  somewhat  more  into  detail  concerning  the  arith 
metical  definition  of  the  definite  integral  of  a  function  of  a  real 
variable.  Let  (a,  b)  be  an  interval,  and  let  a  function  f(x)  be 
given  on  it.  Divide  this  interval  into  any  number  of  subinter- 
vals  by  the  points  xlt  JC2,  •••  xn,*  let  Mv  represent  the  upper 
bound  of  the  values  of  the  function  belonging  to  each  of  these 
subintervals,  and  form  the  sum  : 


(  i  )  M0(x,  -  a}  +  M,(x,  -  A-0  +  M2(xt  -  *,)  + 

(xn  -  *m_0  +  Mn(b  -  x 


This  sum  has  different  values  according  to  the  choice  of  the 
points  determining  the  partition.  But  when  the  values  which 
the  function  takes  on  on  the  given  interval  all  lie  between  two 

*  That  is,  let  x§  =  a,  x±,  x.2<  •••  xn+i  =  b  be  a  set  of  points  lying  in  order  from  a 
to  b.  Such  a  set  of  points  is  called  a  partition  of  the  interval  (a,  t>).  The  intervals 
(jek,  xk+l)  (k  =  i,  2,  •••  «)  are  intervals  of  (a,  6).  —  S.  E.  R. 


152  III.    THE  THEORY   OF   REAL  VARIABLES 

finite  limits  m  and  M,  then  all  possible  values  of  the  sum  (i) 
lie  on  the  finite  interval  \m(b  —  a),  M(b  —  a}~\  and  therefore 
have  a  lower  bound  according  to  II,  §  23. 

I.  This  lower  bound  of  the  values  of  the  sum  (/)  is  called  the 
upper  integral*  of  the  function  f(x]  between  the  limits  a  and  b. 

Under  the  same  assumptions  there  is  an  upper  bound  to  the 
values  of  the  sum 

(2)  mQ(x1  —  a)  +  m^Xz  —  Xi)  +  M2(x3  —  x2)  -f 

+  Mn-i(xn  -  *n-i)  +  mn(b  -  xn), 

in  which  mv  designates  the  lower  bound  of  the  values  of  the 
function  on  the  interval  (xv1  xv+i). 

II.  This  upper  bound  is  called  the  lower  integral  of  f(x)  be 
tween  the  limits  a  and  b. 

No  value  of  (2)  is  greater  than  any  value  of  (i)  even  when 
intermediate  points  are  used  for  the  formation  of  (2)  other  than 
those  used  for  (i) ;  we  see  this  by  further  partitioning  every 
subinterval  used  for  (i)  by  the  points  used  for  (2).  Thus  the 
lower  integral  cannot  be  greater  than  the  upper  integral,  but  at 
most  equal  to  it. 

III.  When  the  upper  integral  is  equal  to  the  lower,  we  call  their 
common  value  simply  the  integral  of  f(x]  between  a  and  b  ;  and  the 
function  f(x)  is  then  said  to  be  integrable  on  the  interval  (a,  b}. 

But  this  is  always  the  case  if  f(x)  is  continuous  on  the  inter 
val.  For  then  according  to  I,  §  24,  for  every  assigned  number 
e  >  o  another,  8,  can  be  so  determined  that,  for  any  two  points 

*  The  terms  upper  integral  (pberes  integral}  and  lower  integral  (unteres  inte 
gral}  were  introduced  by  DARBOUX,  Annales  de  Pecole  normale,  ser.  2,  Vol.  IV, 
and  also  by  THOMAE,  Einleitung,  etc.,  p.  12.  JORDAN,  Cours  d' Analyse,  Vol.  I, 
p.  34,  called  them  "  1'integrale  par  exces  "  and  "  1'integrale  par  defaut."  —  S.  E.  R. 


§28.    INTEGRATION  153 

xlt  x*  of  the  interval, 

\/(xz)  -/(A-0  |  <  -?—  whenever    *,-*>  |  <  8. 
b  —  a 

But  then 
(3) 

whenever  j  xv+1  —  A\,  |  <  8.  If  the  points  of  partition  are  there 
fore  chosen  so  that  these  inequalities  hold  for  each  subinterval, 
we  obtain  two  values  of  the  sums  (i)  and  (2)  which  are  differ 
ent  from  each  other  by  e  at  most.  But  that  would  not  be  possi 
ble  if  the  upper  bound  of  the  smaller  sum  was  different  from 
the  lower  bound  of  the  larger  sum  by  more  than  e.  Since  this 
is  true  for  any  value  of  e,  these  two  bounds  must  be  equal  to 
each  other  (A.  A.  Cor.  to  II,  §  39).  We  have  thus  proved  the 
theorem  : 

IV.  A  function  is  integrable  on  every  interval  on  which  it  is 
continuous. 

It  may  be  mentioned  here  without  proving,  that  the  converse 
of  this  theorem  does  not  hold. 

The  following  theorem  also  arises  from  the  same  proof  : 

V.  If  f(x]  is  integrable  on    the  interval  (a,  b],  then  for  each 
assigned  degree  of  approximation  e  we  can  determine  another  •,  8,  so 
that  the  difference  between  the  value  of  the  sum 


and  the  value  of  the  integral 
(5) 


is  less  than  i(b  —  a),  however  the  subintervals  (xvt  xv+^)  and  on 
them  the  intermediate  values  £v  may  be  chosen,  provided  only  that 
each  of  tJiese  subintervals  is  smaller  than  8. 


154  IIL    THE  THEORY   OF   REAL   VARIABLES 

There  arises  thus  the  possibility  of  computing  the  integral 
of  a  continuous  function  to  an  arbitrary  approximation  pre- 
assigned. 

The  elementary  theorems  about  the  integral  of  a  sum,  etc., 
about  partitions  of  the  interval  of  integration,  about  the  intro 
duction  of  a  new  variable  of  integration  all  follow  without  fun 
damental  difficulties  from  the  definition  of  an  integral  used 
here. 

If  a,  one  of  the  two  limits  of  integration  of  a  continuous 
function,  is  kept  fixed,  while  the  other,  b,  is  considered  as  a 
variable  and  as  such  denoted  by  jc,  then  the  value  of  the  in 
tegral  appears  as  a  function  of  this  variable  ;  let  this  function 
be  denoted  by  F(x).  If  m  and  M  are  upper  and  lower  bounds 
of  the  values  of  the  function  f  on  the  interval  (x,  x-{-  ti),  then 

F(x  +  h)  —  F(x)  =  j      /(£X£  lies  between  mh  and  Mh  ;  hence 

(6)  ,«<^  +  /')-^)<J/, 

h 

and  from  this  it  follows  in  any  case  that 

(7)  \imF(x  +  K)  =  F(x) 

A==0 

and  also,  on  account  of  A.  A.  IV,  §  39  when/(^)  is  in  addition 
to  this  continuous,  that 

(8)  ,u»  *(*  +  *)-*(*)  =f(x) , 

that  is, 

VI.  The  value  of  the  integral  of  a  continuous  function  is  a  con 
tinuous  and,  when  the  integrand  is  continuous,  also  a  differentiable 
function  of  its  upper  limit;  and,  in  fact,  its  derivative  is  in  the 
latter  case  equal  to  the  given  function  itself. 


§  28.    INTEGRATION  I  5  5 

Differentiation  and  integration  are  thus  reciprocal  operations. 

Therefore  methods  for  the  integration  of  rational  integral 
functions  or  of  functions  represented  by  convergent  power 
series  are  deduced  by  reversing  the  corresponding  formulas  for 
differentiation  ;  these  too  are  supposed  to  be  known  here. 

The  following  theorem  now  enables  us  to  obtain  the  integrals 
of  more  complicated  functions. 

VII.    If  on  the  interval  (a,  H) 
(a)  lim/n(,v)  =/(x)  uniformly  as  to  x, 

n===oo 

then  is 

(10)  lira 


For,  hypothesis  (9)  about  the  uniformity  of  approaching  the 
limit  means  that,  for  every  e  we  can  find  an  JV  such  that  for 
every  x  on  the  interval 

(  1  1  )  I/O)  -/„(*)  i  <  e  where  ;;  >  N. 

But  by  one  of  the  elementary  methods  concerning  integration 

just  mentioned,  the  integral  of  a  difference  is  equal  to  the  differ 

ence  of  the  integrals  of  minuend  and  subtrahend  and  the  abso 

lute  value  of  an  integral  is  at  most  equal  to  the  integral  of  the 

absolute  value  of  the  integrand;  it  follows  therefore  from  (n) 

that 

(12)    I  C  f(x)dx  —  \  fn(x]dx    <  i(b  —  a)  whenever  ;/  >  N. 

Since  e(£  —  a)  becomes  arbitrarily  small  as  e  =  o,  the  proof  of 
equation  (10)  is  complete. 

VIII.  In  particular,  an  infinite  series  which  is  uniformly  con 
vergent  can  be  integrated  term  by  term. 

There  are  no  corresponding  theorems  for  differentiation  : 
from  the  hypothesis  alone  that  on  a  given  interval  the  absolute 


156  III.    THE  THEORY  OF   REAL   VARIABLES 

value  of  a  function  remains  less  than  a  given  number,  nothing 
can  be  concluded  as  to  the  value  of  its  derivative  on  this  inter 
val.  But  by  the  application  of  VII  and  VIII  to  the  functions 

JJL  W  we  can  at  least  obtain  the  two  following  theorems  : 
dx 

IX.    If  in  the  neighborhood  of  a  given  point  x 
Km/.(*)  =/(*), 


IF,  FURTHER,    THE  FUNCTIONS    ue        ARE    CONTINUOUS  AND 

dx 

APPROACH    UNIFORMLY    TO    A    DEFINITE    FUNCTION    IN    THE 

LIMIT  AS  N  INCREASES,  thenf(x)  has  a  definite  derivative  at  that 
point  which  is  equal  to  this  definite  function. 

X.  A  convergent  series  of  functions  with  continuous  derivatives 
may  be  differentiated  term  by  term,  WHEN  THE  SERIES  so  FORMED 

IS  UNIFORMLY  CONVERGENT. 

The  extension  of  Theorems  VII—  X  to  the  case  where  the 
general  limiting  process  is  employed  (A.  A.  §  62)  presents  no 
new  difficulties. 

§  29.    Curvilinear  Integrals 

I.  If  in  the  plane  a  path  C  from  a  point  whose  abscissa  is  a  to 
a  point  whose  abscissa  is  b  is  given  by  the  mono  tonic  and  continuous 
function 

(i)  y=f(*)> 

and  if  there  is  also  given  a  function  P(x,  y),  continuous  at  least  along 
this  path,  then  we  shall  understand  the  CURVILINEAR  INTEGRAL. 


(2)  f  />(*, 

*S(C) 

along  the  path  C  to  be  the  integral 

(3) 


§29.    CURVILINEAR   INTEGRALS  157 

II.  A  c  unilinear  integral  changes  its  sign  if  we  change  the 
sense  of  tlie  direction  in  which  tfie  path  is  taken. 

Similarly  (  *  Q(x,  y)dy  is  defined  ;  *  instead  of  \Pdx  +  (  Qdy 
we  may  write  more  briefly  I  (Pdx  +  Qdy)* 

The  curvilinear  integral  along  any  path  (XIV,  §  25)  can 
therefore  be  defined  as  that  integral  which  is  equal  to  the  sum 
of  its  values  along  the  separate  parts  into  which  the  path  is 
divided,  provided  that  for  each  of  these  parts  y  is  a  monotonic 
function  of  x  and  x  is  a  monotonic  function  of  y. 

If  the  functions  x  =  <£(/),  y  =  *f'(f)  defining  the  path  are  dif- 
ferentiable,  then 

(4)  (Pdx  +  Qdy)  = 


But  if   this  is  not  the  case,  the  curvilinear  integral  can  be 
computed  to  an  arbitrary  approximation  by  a  summation  of  the 
form  : 
(5) 


The  following  theorem,  as  also  Theorems  VII,  VIII,  §  28  are 
valid  for  curvilinear  integrals  : 

III.  If  the  functions  P,  Q  are  continuous  in  a  domain  of  the 
plane  and  if,  in  this  domain,  there  is  given  a  set  of  paths  Cn 
which  approach  UNIFORML  V  to  a  definite  path  C  of  tJie  domain  as 
n  increases,  then  is 

(6)  lira  f  (Pdx  +  Q<fy)  =  f  (Pdx  +  Qdy). 

n±x>J(cn)  *Ac) 

The  proof  rests  upon  the  fact  that  P\_x,f(x)~\dx  is  a  contin 
uous  function  of  x  when  P(x,  y)  is  a  continuous  function  of  x 

*  It  is  only  necessary  to  assume  /  monotonic  here  in  order  that  x  may  be  a 
single-valued  function  of  y. 


158  III.    THE  THEORY  OF   REAL  VARIABLES 

and  y,  and  f(x)  is  a  continuous  function  of  x,  and  upon  VII, 
§  28. 

IV.  And,  if  C=  Urn  Cn  is  not  a  proper  but  an  improper  path, 
we  conclude  from  the  premises  that  the  limit  on  the  left  side  of  (6) 
exists  and  that  it  depends  only  upon  C  and  not  upon  the  set  of 
cuives  Cn  used  to  approximate  to  C.     It  can  therefore  be  used  to 
DEFINE  the  curvilinear  integral  for  this  case. 

V.  In  particular,  we  can  approximate  uniformly  and  arbitra 
rily   to   any  path  of  integration  by  a  so-called  "  RECTANGULAR 
CONTOUR"  *  that  is,  by  a  path  composed  of  straight  line  segments 
which  are  alternately  parallel  to  each  of  the  coordinate  axes. 

For,  i  :  If  y  is  a  continuous  and  monotonic  function  of  x 
along  the  path  and  conversely,  and  if  the  degree  of  approxima 
tion  e  is  preassigned,  we  can  then  find  a  finite  number  of  points 
xv,  yv  on  the  path  such  that  none  of  the  differences  |  xv+l  — xv  |, 
\yv+i  —  yv  is  greater  than  c/V2.  Then,  on  account  of  the  pre 
supposed  monotony,  the  piece  (v  •••  v  +  i)  of  the  path  lies  en 
tirely  within  the  rectangle  whose  vertices  are  the  four  points 

(x*  yv\  (xv+*  yv)>  Owi>  ->v+i)>  (x*>  ->wO ;  and  none  of  the 

points  of  the  path  are  more  than  the  distance  e  from  any  one 
point  of  the  sides  of  the  rectangle.  Hence  the  part  of  the 
curve  connecting  (xv,  yv)  and  (#„+!,  yv+i)  can  be  replaced, 
with  an  approximation  e,  by  a  pair  of  intersecting  sides  of  this 
rectangle. 

2.  If  the  path  is  a  proper  one  (XIV,  §  25),  we  can  divide 
it  into  a  finite  number  of  pieces,  for  each  of  which  the  hypothe 
ses  of  the  first  case  above  are  fulfilled. 

3.  If,  finally,  the  path  is  improper,  it  can  be  replaced  with  an 
approximation  e/2,  by  a  proper  path  which  is  then  replaceable, 
with  an  approximation  e/2,  by  a  "  rectangular  contour."     There- 

*  In  German  "  Treppenweg."  —  S.  E.  R. 


§29.    CURVILINEAR   INTEGRALS  159 

fore  this  improper  path  is  replaceable  by  a  "  rectangular  con 
tour  "  with  an  approximation  e. 

It    is   also   possible  to   approximate   to  a    given    curve  by  a 
"  rectangular  contour  "  whose  vertices  have  rational  coordinates. 

VI.    If  Pdx  +  Qdy  is  the  total  differential  dF  of  a  uniform  and 
continuous  function  F(x,  }')  in  the  domain  under  consideration,  then 


(7)  Pdx  +  C//v)  =  F(x»  j-0  -  F(xQ,  jr0), 


however  the  path  of  integration  from  (xc.  JTO)  to  (x^  }\)  is 
chosen  in  this  domain.  This  is  evident  at  once  if  <£(/),  ^(/) 
are  differentiate  along  the  path  ;  for  then,  by  introducing  /  as 
variable  of  integration,  the  integrand  on  the  left  side  of  (7) 

becomes:         dF    dx  dF    d\      ,.     dF     ,. 

—  •  —  •  at  -\  --  -  •  ---  •  at  =  —  •  at. 
dx      dt  dv      dt  dt 

The  same  result  is  obtained  for  other  paths  by  means  of 
Theorems  III  and  IV. 

In  the  general  case,  on  the  contrary,  the  value  of  the  curvi 
linear  integral,  taken  along  a  path  connecting  two  given  points 
of  the  plane,  depends  not  only  upon  those  two  points,  but  essen 
tially  upon  the  path,  since  two  paths  which  connect  the  same 
two  points  give,  in  general,  different  values  of  the  integral.  In 
particular,  the  value  of  the  integral  taken  along  a  closed  path 
is  not  necessarily  zero.  For  our  purpose  it  is  not  necessary  to 
investigate  the  most  general  conditions  under  which  this  oc 
curs  ;  it  is  sufficient  to  deduce  the  following  theorems  : 

If  we  connect  two  points  BD  of  a  closed  path  of  integration 
A  BCD  A  by  a  path  BED  which  does  not  intersect  the  first 
one,  we  obtain  two  closed  paths  of  integration  ABED  A  and 
BCDEB.  Then,  in  general, 

f    =  /  +  /  .  f     =  /  +  f 

JABEDA      J  BED      J  DAB    JBCDEB      J  BCD      J  DB* 


l6o  III.    THE  THEORY   OF    REAL  VARIABLES 


FIG.  15 

Repetition  of  this  result  gives  the  following  theorem : 

VII.  If  a  domain  is  divided  by  paths  into  an  arbitrary  number 
cf  sub  domains,  then  a  given  integral,  taken  along  the  boundary  of 
the  entire  domain,  is  equal  to  the  sum  of  the  corresponding  integrals 
taken  in  the  same  direction  along  the  boundaries  of  the   separate 
subdomains. 

From  this  we  conclude  at  once  that  if  the  integral  is  zero  for 
every  closed  path  of  integration  sufficiently  small  within  a 
simply  connected  domain,*  it  is  also  zero  for  any  closed  path 
of  integration  which  lies  entirely  within  this  domain.  How 
ever,  the  following  theorem  is  more  important : 

VIII.  If  an  integral,   taken  along  the  boundary  of  any  square 
the  length  of  whose  side  is  8  and  which  lies  entirely  within  a  do 
main  B,  is  smaller  than  82e  for  8  sufficiently  small,  then  it  is  zero 
for  every  closed  path  of  integration  belonging  entirely  to  this  do 
main  (where  e  is  understood  to  be  a  number  independent  of  the  selec 
tion  of  this  square,  and  which  approaches  zero  as  8  approaches 
zero). 

*  The  hypothesis  of  simple  connectivity  cannot,  as  a  matter  of  fact,  be  dis 
pensed  with  here;  the  curves  given  as  examples  in  XVII,  $  25  cannot  be  replaced 
by  paths  of  integration  arbitrarily  small  if  they  traverse  the  entire  ring  between  two 
concentric  circles. 


§  29.    CURVILINEAR   INTEGRALS  l6l 

For,  if  we  have  given  a  domain  bounded  by  a  closed  "  rec 
tangular  contour,"  which  may  be  entirely  filled  in  by  such 
squares,  the  value  of  the  integral  around  it  is  smaller  than 
e^Sl  But  282  is  the  surface  F  inclosed  by  the  "  rectangular 
contour  "  and  is  therefore  a  definite  number.  The  value  of  the 
integral  must  then  be  smaller  than  the  product  of  this  number  F 
and  a  number  e  which  can  be  taken  arbitrarily  small.  That  is 
only  possible  when  it  is  zero. 

The  same  theorem  is  valid  for  any  other  arbitrary  closed  path 
of  integration,  as  we  perceive  by  approximating  to  it  by  a  "  rec 
tangular  contour  "  whose  vertices  have  rational  coordinates. 

But  in  the  application  of  this  theorem  it  is  not  convenient  to 
suppose  e  independent  of  the  selection  of  the  square. 

IX.  But  the  same  result  is  obtained  by  supposing  that  to  any 
point  of  B  there  belongs  an  e  such  that  the  conditions  of  VIII  are 
satisfied  for  every  square  belonging  to  tlie  tieighborhood  of  this 
point.  , 

For,  suppose  the  integral  around  any  such  a  domain  were  in 
absolute  value  >  A  ;  we  then  divide  the  path  into  two  parts  ; 

for  at  least  one  of  these  the  integral  must  therefore  be  >  — .  We 
divide  this  one  again  ;  for  at  least  one  of  the  new  parts  the 

integral  must  then  be  >  — .     Continuing  thus  we  arrive  after 

4 

n  divisions  at  a  domain  of  the  surface  F/2*  for  whose  boundary 
that  integral  would  be  >  A/ 2 n.  But  we  can  carry  the  division 
so  far  that  one  of  the  subdomains  thus  obtained,  and  all  the 
following  ones,  would  belong  entirely  to  the  neighborhood  of 
one  of  its  points  ;  for,  the  set  of  its  vertices  must  have  at  least 
one  limit  point.  For  one  such  subdomain,  the  integral  along 
the  boundary  would  therefore  on  the  one  hand  be  <  tF/2n  and 
on  the  other  hand  >  A/2n.  But  this  leads  to  a  contradiction, 


1 62 


III.    THE  THEORY   OF   REAL   VARIABLES 


since  c  can  be  chosen  arbitrarily  small ;  the  contradiction  dis 
appears  only  for  A  —  o.* 


MISCELLANEOUS  EXAMPLES 
1.    Discuss  the  sets  P=  f,  f,  J,  f  ...  ;  that  is, 


« 


p  =  i>  f  >  i»  f >  i>  I  ••• ;  that  is>  p '  5"*Lil  n  >  I 

\ji        n     J 

and  integral,  as  to  upper  (lower)  bound,  limit  points,  superior 
(inferior)  limit  L  (Z),  derived  sets,  and  whether  dense,  closed. 

2.  The  positive  rational  numbers  can  be  arranged  in  the  form 
of  a  simple  sequence  as  follows : 

l,i  M,  f>M>  iii-- 

Show  that  //^  is  the     -  (/  -f  ^  —  i)(/  +  q  —  2)  +  ^  r  term 
of  the  series. 

Discuss  for  continuity  the  functions : 

3.  y  =  \/xi  at  x  =  o. 

\y  y 


—   I 


4.    y  =  -  —  at  x  =  o.          5.    y  =  i/x  at  x  =  o. 


*  For  this  and  many  similar  arguments  the  HEINE-BOREL  theorem  is  of  direct 
use.  For  an  exposition  of  this  theorem  see  VEBLEN  AND  LENNES,  /.  c.,  p.  34. 
—  S.E.R. 


§  29-    CURVILINEAR  INTEGRALS 


163 


6.  v  =  sin  -  at  x  =  o.     In  this  case  discuss  also  the  oscilla- 

x 

tions  of  y  and  find  its  value  as  x  =  o  from  the  left ;  from  the 
right. 

7.  y  =  x  -  sin  -  at  x  =  o.     Discuss  as  in  Ex.  6. 

x 

8.  y=  —  +  ~r-sin-  at  x  =  o.     Here  v  oscillates  about  the 

JC2  X 

curve  }'=i/x2.     Show  that  the  amplitude  of  these  oscillations 
converges  to  zero  as  x=o. 

9.  v  =  -  •  sin  -  at  x  =  o.     Here  y  oscillates  between  the  two 

x          x 

hyperbolas  y  =  ±  i  /x.     Show  that  as  x  =  o  the  amplitude  of 
these  oscillations  increases  indefinitely. 


O 


10.    Let  y  =  i  for  x  =£  o  and 

=  o  for  x  =  o.     The  analytic  expression  of  y  is  then 

'     nx 


In  what  respect   is  the  disconti 


nuity  in  this  case  different  from  that  in  the  other  examples  just 
studied  ? 

The  two  following  are  examples  of  continuous  functions  for 
which  progressive  or  regressive  derivatives  at  certain  points  do 
not  exist. 


1 64 


III.    THE  THEORY   OF   REAL  VARIABLES 


11.    Let 


y  =  x  •  sin  —  for  x  =£  o  and 
x 

=  o  for  x  =  o. 


Here  y  oscillates  infinitely  often  between  the  lines  y  =  ±  x  as 
x  =  o.     Now  for  x  •=£  o,  y  is  certainly  continuous  ;  but  it  is  also 

/  7T\ 

continuous  for  x  =  o  since    lim  (  x  •  sin  -  )  =  o.     At  the  origin 


there  is  no  tangent  at  all  to  the  curve  since  a  secant  at  the 
origin  oscillates  between  the  two  lines  and  approaches  no  fixed 
position. 

Analytically,  this  is  shown  as  follows : 


As  AJC  =  O,  sin  —  -  oscillates  infinitely  often  between  ±  i.     Cf. 
Ex.  6. 

y 


12.    Let 


y  =  x2  •  sin  —  for  x  =£  o  and 
=  o  for  x  —  o. 


Here  y  is  continuous  everywhere,  even  at  x  =  o,  and  oscillates 
between  the  two  parabolas  y  —  ±  x*  and  increasingly  often  as 
X  =  Q.  As  a  point  on  the  curve  approaches  zero,  a  secant 
through  this  point  and  the  origin  oscillates  between  narrower 


§  29.    CURVILINEAR   INTEGRALS  165 

and   narrower    limits.     These    limits    converge    on    both    sides 
toward  the  .r-axis.     The  tangent  therefore  at  the  origin  is  the 


TT      I 

A.v  •  sm  —    =  o. 


Analytically  : 


Al'  •          7T  IT 

-*-  =  A.v  -sin  —  -  at  x  =  o  and  hm 

A.#  Ajf  Ari<)_ 

13.  The  function  defined  by  /(x)  =  x]  i  +  -  sin  (log  x~)  \   and 

«3 

/(o)  =  o  is  everywhere  continuous  and  monotonic. 
[PRINGSHEIM,  Encyklopadie  der  Math.  W  is  sen..  II  A.  i,  p.  22.] 

Investigate  whether  this  function  has  at  x  =  o  a  progressive 
or  a  regressive  derivative  or  both,  and,  if  both  exist,  whether 
they  are  the  same. 

14.  Discuss  Exs.  3-9  for  progressive  and  regressive  deriva 
tives  as  in  Ex.   13. 


15.  Given  a  rational  function  of  A,  r(x)  =  where  g(x) 

and  h(x)  are  polynomials.     Show  that  in  no  case  can  the  de 
nominator  of  /(A*)  be  a  simple  factor  as  (x  —  a). 

Hence  show  that  no  rational  function  (such  as  i/x)  whose 
denominator  contains  any  simple  factor  can  be  the  derivative 
of  another  rational  function. 

16.  The  functions  it,  z>,  of  x  and  their  derivatives  ?/,  v'  are 
continuous  throughout  a  certain  interval  of  values  of  x,  and  //?'' 
—  u'v  never  vanishes  at  any  point  of  the  interval.     Show  that 
between  any  two  roots  of  //  =  o  occurs  one  of  v  =  o,  and  con 
versely. 

[If  v  does  not  vanish  between  two  roots  of  u  =  o,  say  a  and  /3,  the  func 
tion  u/v  is  continuous  throughout  the  interval  (a,  /3)  and  vanishes  at  its  ex 
tremities.  Hence  (ujv)[  —  (u'v  —  uv'^/v2  must  vanish  between  a  and  ft 
which  contradicts  the  hypotheses.] 


1 66 


III.    THE  THEORY   OF   REAL  VARIABLES 


17.  The    constituents  of    an    nth   order    determinant   A    are 
functions   of   x.       Show  that   its    derivative    is   the  sum    of   n 
determinants  each  of  which  is  obtained  from  A  by  substituting 
the  derivatives  of  the  elements  of  a  row  for  the  elements  them 
selves. 

18.  If  /i,/2,/3,/4  are  polynomials  of  degree  not  greater  than 
4,  then 


fin       -f>»       -fin       -ff>f 
J\          /2          /3          /4 

is  also  a  polynomial  of  degree  not  greater  than  4.  [Differenti 
ate  five  times,  using  the  result  of  Ex.  17  and  rejecting  vanishing 
determinants.] 

19.    If  /(x),  <j>(x),  \l/(x)  have  derivatives  for  a  <  x  ^  b,  there 
is  a  value  of  £,  lying  between  a  and  b  and  such  that 


/(a) 


=  o. 


[Consider  the  function  formed  by  replacing  the  constituents 
of  the  third  row 


CHAPTER    IV 

SINGLE-VALUED  ANALYTIC   FUNCTIONS   OF  A  COMPLEX 
VARIABLE 

§  30.    Introduction 

WE  have  already  introduced  and  investigated  in  part  in 
Chapter  II  a  series  of  elementary7  functions  of  a  complex  vari 
able  2.  But  at  that  time  we  postponed  the  discussion  of  the 
concept,  "Function  of  a  Complex  Variable  ";  this  will  now  be 
considered. 

We  can,  to  be  sure,  call  X+iY  in  the  most  general  sense 
a  function  of  x  +  iy  if  the  real  expressions  X,  Y  are  func 
tions  of  the  real  variables  x  and  y.  The  theory  of  functions 
of  a  complex  variable  would  then  be  nothing  else  than  the 
theory  of  pairs  of  functions  of  two  real  variables.  It  is, 
however,  customary  to  use  the  word  in  a  narrower  sense, 
so  that  the  "  Theory  of  Functions  of  a  Complex  Variable  " 
represents  only  a  particularly  important  and  interesting 
chapter  in  the  theory  of  pairs  of  functions  of  two  real  varia 
bles.  The  following  considerations  form  a  basis  for  this  point 
of  view : 

The  particular  phrase,  rational  function  of  a  complex  quantity 
x  +  ty,  has  already  been  given  a  definite  meaning  in  Chapter  II 
on  the  basis  of  the  definition  of  the  elementary  operations  with 
complex  quantities  given  in  Chapter  I.  One  might  now  be 
tempted  to  take  as  the  basis  for  the  definition  of  a  transcen 
dental  function  of  a  complex  argument  that  definition  of  a 

167 


1 68  IV.    SINGLE- VALUED   ANALYTIC   FUNCTIONS 

transcendental  real  function  based  on  limits  of  rational  func 
tions —  for  example, 


But  there  are  serious  objections  to  this :  it  may  happen  that 
such  limits  exist  for  all  real  but  for  no  complex  values  of  x ;  it 
may  further  happen  that  two  such  limits,  which  represent  the 
same  transcendental  function  when  x  is  real,  are  different  when 
x  is  complex.  But,  as  a  matter  of  fact,  we  shall  soon  see  (§  38) 
that  such  difficulties  do  not  appear  in  a  certain  class  of  such 
functions,  that  is,  for  sums  of  infinite  power  series.  Accord 
ingly,  WEIERSTRASS*  took  the  theory  of  power  series  as  the 
basis  of  his  theory  of  functions.  CAUCHY  and  RIEMANN,  on  the 
contrary,  began  in  general,  not  with  an  analytical  expression,  but 
with  a  definite  property  which  belongs  to  every  rational  function 
of  a  complex  variable  but  not  to  every  expression  X  +  iY  whose 
members  are  rational  functions  of  the  real  variables  x,  y.  We 
shall  follow  the  latter  point  of  view  here.  It  is  essential  there 
fore  that  we  become  acquainted  with  this  distinctive  property 
of  rational  functions  of  a  complex  variable  ;  the  following  para 
graphs  are  a  preparation  to  that  end. 

§  30  a.   Limits  of  Convergent  Sequences  of  Complex  Numbers 

The  application  of  the  conception  of  a  convergent  sequence 
of  numbers  to  complex  numbers  raises  no  essential  difficulties. 
For,  if 

then  (cf.  also  I,  §  25) 

x    <  e  and    y    <  e. 

*  An  authentic  publication  of  the  lectures  of  WEIERSTRASS  has  been  promised 
for  years.  The  work  by  J.  THOMAE,  Elementare  Theorle  der  analytischen  Funkti- 
onen  einer  complexen  Ver  Underlie  hen  (Halle,  1880,  2d  ed.  1898),  and  that  of  Ch. 
MERAY,  Lemons  nouvelles  sur  I' analyse  infinitesimale  (Paris,  1894- 1895), are  written 
from  the  same  point  of  view. 


§30  a.    LIMITS   OF  CONVERGENT   SEQUENCES  169 

If  therefore  a  sequence  of  complex  numbers 

ZQ=XQ  +  O'j,  Z1  =  Xl  +  />!,  Z,  =  A"o  +  M'2,    —   Zn  =  Xn  +  *>„,   ••• 

is  so  arranged  that  for  every  given  degree  of  approximation  £ 
we  can  so  determine  an  integer  «  that 
(0  |  ;,l+p  -s,,!<c 

for  ever}"/  >  o,  then 

(2)  |  *n+.p  -  *n  !   <   C>    !.)Wp  "A  I    <   C 

for  the  same  value  of  the  integer  ;/  and  for  every/  >  o.  There 
fore  the  real  and  the  pure  imaginary  parts  of  z  form  convergent 
sequences  of  numbers  and  the  limits 

(3)  lim  >*n  —  a<>    lim  }'n  —  ^ 

nix  »i=^» 

exist.  Conversely,  if  inequalities  (2)  exist  for  a  definite  «  and 
all  values  of  /  >  o,  then  the  following  inequality 

(4)  \Zn+p-Zn\<^2 

exists  under  the  same  conditions.  By  putting  a  -f-  ib  =  c  we 
may  combine  the  two  limits  (3)  into  one  and  define  : 

I.  The  limit  lim  zn  =  c 

n=x 

shall  be  taken  to  signify  the  system  of  equations  (j). 

We    extend  this  at  once  and  write  the  definition  (cf.   A.  A. 

§53): 

II.  An  infinite  series  of  complex  quantities 

SQ  +  %  +  z,  -f  -  -h  zn  +  ... 

is  called  convergent  and  the  complex  quantity  S  is  called  tJie  sum  of 
the  series,  when  the  limit 


exists  and  is  equal  to  *£. 


I/O  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

The  following  is  another  formulation  of  the  same  definition : 

III.  The  necessary  and  sufficient  condition  for  the  convergence  of 
an   infinite  series  of  complex  quantities  is,  that  the  series  of  real 
parts  and  the  series  of  imaginary  parts  respectively  converge. 

We  define  further : 

IV.  A  series  of  complex  quantities  is  called  absolutely  *  conver 
gent  when  the  series  formed  by  its  absolute  values  converges. 

On  the  basis  of  III,  §  5  of  this  text  and  §  40,  A.  A.,  we  then 
have  the  following  theorem  : 

V.  If  a  series  of  complex  quantities  converges  absolutely,  then  the 
series  formed  respectively  from  its  real  parts  and  from  its  imagin 
ary  parts  converge  absolutely  ; 

and  from  this,  on  the  basis  of  I,  §  58,  A.  A.,  we  state  the  more 
general  theorem  that : 

VI.  The  sum  of  an  absolutely  convergent  series  of  complex  quan 
tities  is  independent  of  the  arrangement  of  the  terms. 

§  31.    Continuity  of  Rational  Functions  of  a  Complex  Variable 

For  the  sake  of  completeness  we  begin  with  the  definition  : 

I.  A  complex  function  of  one  or  more  real  variables  is  a  complex 
variable  Z  =  X  +  i  Y  whose  components  X,  Y  are  functions  of  those 
variables. 

If  there  are  two  independent  variables,  say  x,  y,  we  can  com 
bine  them  as  one  complex  variable  z  =  x  -f  iy  and  write 

(i)  Z=f(z). 

As  a  matter  of  fact  we  shall  do  this  provisionally ;  later  this 
terminology  will  be  used  only  in  a  more  restricted  sense. 

*  Also  called  unconditionally  convergent.  The  terms  absolutely  convergent  and 
unconditionally  convergent  are  co-extensive, —  S,  E,  R, 


§31.    CONTINUITY  OF   RATIONAL  FUNCTIONS  171 

The  conception  of  continuity  (§  24;  A.  A.  §  61)  is  applicable 
directly  to  complex  functions  on  the  basis  of  the  definitions  of 
§  3°^- 

II.    A  complex  function  is  called  continuous  at  a  definite  point 
z  =  a  if  the  equation 
(2)  lim/(sj  =f(a) 


exists  for  EVERY  sequence  of  numbers  for  which 

(3)  lim  zn  =  a. 

n=x 

Comparison  with  §  26  shows  this  definition  to  be  synonymous 
with  the  following  : 

A  function  of  a  complex  variable  z  =  x  +  iy  is  said  to  be  a  con 
tinuous  function  of  z  only  when  it  is,  in  tJie  sense  defined  in  §  26, 
III,  a  continuous  function  of  the  two  real  variables  x  and  y  (not, 
however,  when  it  is  a  continuous  function  of  x  and  a  continu 
ous  function  of  y). 

We  proceed  accordingly  to  apply  the  general  conception  of 
limits  (A.  A.  §  62)  to  complex  functions  ;  thus  : 

III.    The  equation 

(4)  lim/(S)  =  J 

z±a 

means  tJie  same  as 

(5)  lim/fe)  =  b 


for  E  VER  Y  sequence  of  numbers  converging  to  a. 

Therefore,  a  complex  function  is  said  to  have  at  a  certain 
point  a  definite  value  in  the  limit,  only  when  this  value  is 
reached  by  the  use  of  arbitrary  values  of  approximation  for  the 
argument,  or  geometrically,  if  we  approach  the  same  value  of 
the  function  by  allowing  the  argument  to  approach  its  value 
along  any  curve  whatever.  To  be  sure,  we  have  frequently  to 


IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

consider  approaching  a  limiting  point  not  along  arbitrary  curves 
but  only  along  particular  ones,  for  example,  approaching  the 
origin  along  only  such  curves  which  do  not  encircle  it  an  infi 
nite  number  of  times,  or  along  only  such  as  remain  entirely  in 
the  positive  half  of  the  plane.  Such  limitations  must  then  be 
expressly  stated  each  time. 

IV.  The  theorem  that  the  sum,  difference,  product,  and  quotient, 
providing  the  denominator  is  not  zero,  of  two  continuous  functions 
are  themselves  continuous  functions  is  true  for  complex  variables  as 
well  as  for  real. 

For,  the  proof  of  this  theorem  rests  only  upon  the  two 
theorems  (A.  A.  §  64)  that  the  absolute  value  of  a  sum  or  differ 
ence  is  not  greater  than  the  sum  of  the  absolute  values  of  the 
separate  parts,  and  that  the  absolute  value  of  a  product  or  of  a 
quotient  is  equal  respectively  to  the  product  or  the  quotient  of 
their  absolute  values.  But  these  two  theorems  hold  for  com 
plex  numbers  as  well  as  for  real  (III,  §  5  ;  I,  §  6  ;  II,  §  7). 

Since  it  follows  directly  from  definition  II  that  z  itself  is  a 
continuous  function  of  z,  we  obtain  the  theorem  (cf.  IV,  §  26) : 

V.  A  rational  function  of  a  complex  variable  is  everywhere  con 
tinuous  where  it  is  finite. 

We  have  therefore  to  consider  only  the  results  of  §  20  by 
which  a  rational  function  can  always  be  put  in  such  a  form  that 
the  denominator  is  zero  only  where  the  function  is  infinite. 

It  follows  further  from  the  results  of  §  20  that : 

VI.  At  the  poles,  at  which  a  rational  function  itself  is  not  con 
tinuous,  its  reciprocal  at  least  is  continuous. 

Theorems  V  and  VI  are  understood  to  hold  for  finite  values 
of  the  independent  variables ;  but  they  are  valid  also  for  the 
value  oc  according  to  §§  12  and  21  ;  that  is, 


§3i.    CONTINUITY   OF   RATIONAL  FUNCTIONS  1/3 

VII.   Either  a  rational  function  or  its  reciprocal  (or  both)  are 
continuous  for  z  =  oo. 

Equations  of  the  form 
(6)  /(*>)=*,  /(*0  =  «'o,  /(*>)  =  oo 


were  regarded  in  a  purely  conventional  way  in  §§20  and  21 
according  to  the  meaning  given  to  the  symbol  "  oo  "  in  §  12. 
However,  such  equations  can  be  interpreted  in  a  different  way 
(A.  A.  §  63)  which  is  applicable  to  complex  variables  as  fol 
lows  : 

VIII.  /(%)  =  oo   means   that  for  every  given    number  M~>  O 
another  number  8  can  be  determined  such  that 

|  <  8. 

IX.  /(oo)  =  WQ    means   that  for    every  given    number  e  >  o 
another  number  N  can  be  determined  such  that 

\f(z)  —  WQ    <  c,  whenever  \  z  \  >  N. 

X.  /(oo)  =  oo    means    that  for  every  given    number   M  >  o 
another  number  N  can  be  determined  such  that 

I/O)  I  >  M,  whenever  \  z  \  >  N. 

Theorems  VI  and  VII  then  assert  that  : 

XI.  These   two  views  of  the  symbol  oc  as  applied  to  rational 
functions  are  not  contradictory  ;  and  every  such  equation  (6)  which 
is  true  from  the  one  point  of  view  is  also  true  from  the  other. 

Geometrically,  the  theorems  of  this  paragraph  assert  that  : 

XII.  The  map  of  the  z-sphere  upon  the  w-sphe?-e  determined  by  a 
rational  function  w  =f(z)  is   everywhere  continuous,  even  in  the 
vicinity  of  the  point  oo  of  both  spJieres. 


1/4  IV-    SINGLE-  VALUED   ANALYTIC   FUNCTIONS 

Further,  we  are  always  to  understand  that  also  the  second  and 
third  of  equations  (6)  are  valid  when  the  point  of  the  sphere 
representing  the  independent  variable  approaches  the  point  oo 
of  the  sphere  along  any  arbitrary  curve.  When  specially  con 
sidering  such  curves  it  must  be  so  stated  each  time,  as,  for  ex 
ample,  when  it  is  to  be  merely  affirmed  that  a  definite  value  is 
reached  in  the  limit  as  the  independent  variable  increases 
through  positive  real  values  beyond  all  bounds. 

§  32.    Derivative  of  a  Rational  Function  of  a  Complex  Argument 

To  pursue  further  the  investigation  spoken  of  at  the  close  of 
§  30,  we  study  the  quotient 

(i)  ^+_<I 

as  a  function  of  £  and  for  a  definite  value  z0  of  z  for  which  the 
rational  function  f(z)  is  finite.  It  is  a  rational  function  of  £ 

which  takes  the    indeterminate  form    -   for  £  =  o.     But  it  has 

o 

already  been  shown  in  §  20  that  such  an  indeterminate  form  of 
a  rational  function  of  £  can  always  be  evaluated  by  a  suitable 
reduction.  In  other  words,  we  can  always  find  another  rational 
function  ^(£)  which  agrees  with  i//(£)  for  all  those  values  of  £ 
for  which  i//(£)  is  determinate,  but  which  for  £  =  o  either  has  a 
definite  value  or  is  definitely  infinite  in  the  sense  defined  there. 
Now  it  is  shown  in  the  differential  calculus  (cf.  also  §  27)  that, 
restricting  ZQ  and  £  to  real  values,  this  function  ^(£)  under  the 
given  assumptions  does  have  a  definite  value  for  £=  o  and  that 
this  value  is  a  rational  function  of  z0  which  is  designated  by 


and  which  is  customarily  called  the  derivative  of  /(z).     But  in 
this  way/(z)  is  supposed  real  at  the  outset;  however,  the  same 


§32.    DERIVATIVE  OF   A   RATIONAL   FUNCTION          175 

process  is  applicable  here  by  dividing   \j/(£)  into  its  real   and 
imaginary  parts  : 


and  treating  each  of  these  parts  separately.  It  then  follows 
that  the  quotient  ^(£),  under  the  assumption  that  /(ZQ)  =£  oc  for 
£  =  o,  has  a  definite  value  /'(%)  in  the  limit  (and  dependent 
upon  z0)  when  £  is  restricted  to  real  values.  But  we  have  seen 
in  the  preceding  paragraphs  that  a  rational  function  of  a  com 
plex  variable  £  is  everywhere  continuous  where  it  is  finite  ;  if 
therefore 


whenever  £  approaches  zero  through  real  values,  it  follows  for 
rational  functions  /,  that  this  equation  must  hold  ///  whatever 
manner  £  converges  to  zero.  These  results  are  stated  in  the 
following  theorem  : 

I.    A  rational  function  f(z]  of  a  complex  variable  has  at  ei'ery 
point  z  at  which  it  is  finite  a  definite  derivative, 


(3)  =/'(*)' 

independent  of  the  manner  in  which  dz  approaches  zero  and  which 
can  be  found  by  the  methods  of  the  differential  calculus  for  real 
variables  and  functions. 

It  is  now  easy  to  see  that  this  property  does  not  belong  to 
every  expression  //  +  /?'»  whose  members  are  rational  functions 
of  x  and  y.  For,  the  total  variation  of  such  an  expression  is, 
according  to  elementary  theorems  of  the  differential  calculus 
for  functions  of  two  variables, 

.    .A         fdu  ,    .  dv  ,      \  .       ,  /  dit  ,    .  dv  .     \  A 
A//  +  /Az'=    —  +  i  —  +  £l  A,v  +   —  +  /  TT  +  C2  Ay 
\d*         dx         J  \dy         dy         J 


1/6  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

where  ely  £2  designate   quantities  that  approach  zero  with 
and  Ay.     The  quotient 


A#  -f  / 
can  then  be  written 


If  we  now  allow  AJC  and  Ajy  to  approach  zero  in  such  a  way  that 

—  -2-  con-verges  to  a  definite  value*  -^  in  the  limit,  that  is.  gea 

AJC  d& 

metrically,  if  we  allow  the  point  (x  -f-  AJC,  jj'  +  AJK)  to  approach 
the  point  (jc,  jv)  along  a  curve  which  has  a  definite  tangent  at 
C#i  y)>  then  the  above  quotient  converges  in  the  limit  to 


In  general  this  expression  depends  essentially  upon  -*-  in  the 
limit ;  it  will  be  independent  of  -*-  when  and  only  when  the 
term  free  from  -2-  in  the  numerator  (as  in  the  denominator) 

HOC 

has  the  ratio  i  :  /  to  the  coefficient  of  -^,  in  other  words,  when 

dx 

,  ^  du       .dv_  ./du       .  dv\ 

dy         dy        \dx         dx) 

But  this  equation  is  true  (I,  §  2)  when  and  only  when 


(5) 


, 

T-  =  -T-  and 
d_y  o^ 

dv  _       du 
dy  dx 

*  Which  may  also  be  oo. 


§32.    DERIVATIVE  OF  A   RATIONAL  FUNCTION          177 

We  have  thus  the  result  : 

II.  An  expression  of  the  form  it  -\-  iv,  in  which  it,  v,  are  ra 
tional  functions  of  x  and  y,  can  only  be  pit  t  in  the  form  of  a  rational 
function  of  z  =  .r  +  iy  when  u  and  v  satisfy  the  partial  differential 
equations  (5). 

On  the  other  hand,  it  is  to  be  noticed  that  the  formal  rules 
for  the  differentiation  of  rational  functions  are  simple  conse 
quences  of  fundamental  theorems  of  elementary  algebra.  Since 
we  have  shown  in  the  first  chapter  that  these  fundamental 
theorems  hold  for  complex  expressions  as  well  as  for  real,  it 
follows  that  we  may  also  apply  these  rules  of  differentiation 
to  rational  functions  of  a  complex  variable.  Thus,  for  example, 
in  such  a  function,  considered  as  a  function  of  x  and  y,  we 
can  introduce  z  =  x  -f  iy  in  place  of  x  as  independent  variable 
along  with  y  ;  let  us  then  distinguish  the  partial  derivatives  taken 
with  respect  to  these  independent  variables  from  those  taken 
with  respect  to  x  and  y  as  independent  variables  by  inclosing 
them  in  parenthesis.  Thus,  according  to  these  rules  : 


dx      \dz        dy      \d 

If  now  we  have  a  complex  expression  //  -f  jv,  whose  members 
are  rational  functions  of  x  and  y  and  which  satisfies  equation 
(4)  and  if  we  replace  /in  (6)  by  it,  it  follows  that 


When,  therefore,  z  =  x  -\-  iy  is  introduced  in  place  of  x  along 
with  y  as  a  new  independent  variable  in  a  complex  expression 
of  the  given  form  satisfying  equation  (4),  y  itself  drops  out  ;  in 
other  words  : 


1/8  IV.    SINGLE-  VALUED    ANALYTIC   FUNCTIONS 

III.  If  H  and  v  are  rational  functions  of  x  and  y,  then  the  exist 
ence  of  equations  (5)  is  not  only  a  necessary  but  is  also  a  sufficient 
condition  that  u  •+-  iv  can  be  put  in  the  form  of  a  rational  function 
of  z  alone. 

§  33.    Definition  of  Regular  Functions  of  a  Complex  Argument 

The  property  of  rational  functions  of  a  complex  variable  de 
duced  in  the  last  paragraph  will  now  be  taken  as  the  starting 
point  for  the  determination  of  a  general  definition  of  a  function 
of  a  complex  argument  : 

I.  w  =f(z]  shall  be  called  a  (^regular)  function  of  a  complex 
argument  z  in  a  given  domain,  only  when  the  limit 


(i) 

( 

exists  in  the  sense  defined  in  III,  §  ji  for  every  \  point  z  of  this 
domain. 

The  symbol  f(z)  will  be  used  exclusively  hereafter  for  such 
regular  functions  of  z,  and  the  limit  (i)  will  be  designated  by 

-*-±-t-  or/'(z)  just  as  for  real  variables  and  functions.* 
dz 

If  the  function  w=-u-\-iv 

be  separated  into  its  real  and  imaginary  parts  and  if  the  func 
tions  u,  v  have  continuous  partial  derivatives,  then  the  results 
of  §  32  show  that  the  limit  (i)  is  independent  of  the  manner  in 
which  z  approaches  zero  only  when  these  partial  derivatives 
satisfy  equations  (5),  §  32.  Conversely,  reviewing  these  results 
starting  with  the  last,  it  follows  that  these  equations  together 

*  To  indicate   that  a  function  w  =f(z}    has  the  property  that  —    tends,  in 

Az 

general,  to  a  unique  finite  limit,  that  is,  that  it  satisfies  (5),  §  32,  CAUCHY  employed 
the  term  monogenic,  while  RlEMANN  dispensed  with  the  adjective  altogether.  Cf. 
RlEMANN,  Ges.  Werke,  pp.  5,  81.  —  S.  E.  R. 


§33.    DEFINITION  OF   REGULAR   FUNCTIONS  179 

with  the  assumption  of  continuity  of  the  partial  derivatives  ap 
pearing  in  them  are  sufficient  to  infer  the  existence  of  limit  (i) 
in  the  established  sense.  On  this  account  CAUCHY  and 
RIEMANN  made  use  of  these  differential  equations  as  the  def 
inition  of  functions  of  a  complex  argument. 

Besides,  we  notice  that,  in  passing  from  the  differential  equa 
tions  to  the  limit  (i)  and  conversely,  it  was  necessary  to  assume 
the  continuity  of  the  partial  derivatives  which  appeared  ;  on  the 
contrary  we  shall  see  that  in  the  further  application  of  the  limit 
(i),  we  need  only  assume  its  existence  for  each  point  of  the  do 
main,  not  its  continuity  as  a  function  of  z.  This  continuity  fol 
lows  rather  as  a  consequence  of  the  above  assumptions. 

An  example  of  a  regular,  but  not  rational  function  of  a  com 
plex  argument  is  obtained  by  putting 

//  =  e*  cos  y,  and  v  =  f  sin  y  ; 

for,  from  these  equations,  we  find 

du      dv  du          dv 

—  =  —  =  e*  cosy,     —  =  -  —  =  -exsmy. 
dx      dy  By  dx 

Thus,  £*(cos_>'  +  /  s'my)  is  a  function  of  z  =  x  +  /)',  regular  over 
the  whole  plane. 

Definition  I  does  not  in  general  require  the  existence  or  con 
tinuity  of  higher  derivatives  (but  we  shall  see  later  that  they  can 
be  inferred  from  this  definition).  But  if  this  result  is  assumed 
repeated  differentiation  of  the  differential  equations  (5),  §  32, 
leads  to  the  following  results  : 

xv  Cfrt         d^  = 

* 


a/      dxdy      dydx 

,  v  v         v  __   _  d*u         B*u   _ 

~      ~~~'  dydx~ 


180  IV.    SINGLE- VALUED   ANALYTIC   FUNCTIONS 

hence  the  following  theorem  : 

II.  Neither  the  real  nor  the  pure  imaginary  part  of  an  ana 
lytic  function  of  a  complex  argument  can  be  assumed  to  be  arbitrary 
functions  of  x  and  y ;  on  the  contrary,  each  must  satisfy  the  corre 
sponding  "  LAPLACE  "  differential  eqiiation  : 

dht      d2ft  _     _    dzv   .  dzv  _ 
~     '  ~ 


EXAMPLES 

1.  An  example  exhibiting  a  function  whose  derivative  at  a 
point  is  not  independent  of  the  manner  of  approaching  this 
point  is  the  following : 

Let  w  =  ft  -f  iv  =  2  x  -f-  3  iy  be  the  function,  and  let  us  ex 
amine  it  at  a  point  (x,  y).  At  a  neighboring  point  we  obtain 

w  -\-  &.w  =  ft  +  Aft  +  i(v  +  A 
.'.  Aft  =  2  Ax,  Az/  =  3  Ay. 

The  derivative  at  a  point  (x,  y)  therefore  becomes 
lim|^±^l      limr-  A-  '   -'-  A"~' 


The  value  of  this  derivative  can  be  made  to  assume  any  arbi 
trary  value    by  suitably  choosing  -*•  •     This  process  does  not 

dx 

therefore    define   a   derivative    independent  of   the    manner  of 
approaching  the  given  point. 

Show  directly  that  w  =  2  x  -f  3  iy  is  not  a  regular  function. 

2.    If  ft  =  (x  —  i)3  —  3  xyz  -f  3  jy2,  determine  27  so  that  u  +  &  is 
a  regular  function  of  ^  +  (y. 

Ans.    v  =  3  /(jf  —  i)2  —  y,  that  is,  a/  =  (z  —  i)3. 


§33-    DEFINITION   OF   REGULAR   FUNCTIONS  l8l 

3.  If  u  =  x3,  is  it  possible  to  determine  v  so  that  the  function 
is  regular  ? 

4.  Given  v=  2  y(x  +  i) ;  determine  the  corresponding  u  so 
that  u  +  iv  shall  be  a  regular  function. 

5.  Given  //  =  jc3  —  3  x)&  ;  find  the  corresponding  v  as  in  Ex.  3. 

6.  Given  //  =  e*~-y~  •  cos  2  xy ;  find  v  and  the  resulting  func 
tion  of  z  as  in  Ex.  3. 

7.  Prove  by  passing  directly  to  the  limit  that  in  polar  coor 
dinates  the  CAUCHY-RIEMAXN    differential  equations  take   the 
form  : 


'dtt_ 

I    9v 

dr 

r  '  d<t>' 

dv  _ 

i     dtf 

ttt 

.dr 

r     d<f> 

u 
dw       _4>t-     da:* 

e        2 

dz                 dr 

r 

Also,  show  that 

dw 

—  = • 

)r  r 

8.  Show  that  the  function 

w  =  log  r  4-  '<£» 

where  s  =  r(cos  <#>  +  /  sin  <£),  is  regular  at  every  point  z  different 
from  o  and  oo.     (Cf.  §  56.) 

9.  If  a  function 

f(x  +  /v)  =  ^(cos  6  -I-  /  sin  ^) 

is  regular  in  a  definite  domain,  then 

n    73  S/l  n    O  J3Z1 

O/t   „       Ot/  U/[   _  ,p       Ov 

dx  dy'      dy  dx' 


and 


Derive  also  the  corresponding  relations  for  the  case  where 
is  expressed  in  the  form  z  =  r  •  eie. 


1 82  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

§  34.    Conformal  Representation 

We  have  already  investigated  in  §  27  the  mapping  of  a  do 
main  of  the  jcy-plane  continuously  on  a  region  of  the  zw-plane. 
A  particular  class  of  such  transformations  are  those  for  which 
u  -+-  iv  is  a  regular  function  of  x  -f-  iy  in  the  sense  defined  in  the 
previous  paragraph  ;  we  wish  to  characterize  this  class  of  trans 
formations  geometrically. 

For  this  purpose  we  rearrange  the  preceding  results  some 
what.  Let  zlt  z2,  z3  be  three  values  of  z  =  x  -\-iy,  w±,  w2j  ws  the 
corresponding  values  of  w  =  u  -}-  w,  and  form  the  quotients 

and 


Z2  —  Z{  £3  —  Zi 

As  zz  and  %  approach  zit  these  two  quotients  differ  by  an  in 
finitesimal  *  ;  for,  each  of  them  differs  by  an  infinitesimal  from 
the  definite,  unique  value  of  the  derivative  : 

dw 
dz 

at  the  point  z  =  zv     Accordingly, 


where  c  becomes  infinitesimal  with  z^—  Zi  and  z3  —  %.  Con 
versely,  when  such  an  equation  exists,  in  whatever  way  z2  and 
z3  may  approach  the  point  zlt  it  follows  that  the  derivative  —  - 

(t% 

is  independent  of  the  direction  of  the  differential  dz. 

Apart  from  an  exception  to  be  spoken  of  presently  (VI),  we 
can  draw  the  general  conclusion  from  equation  (i),  that  also  in 

*  We  shall  understand  that  no  constant,  however  small,  if  not  zero,  is  an  infini 
tesimal  ;  the  essence  of  the  infinitesimal  is  that  it  varies  so  as  to  approach  zero  as 
a  limit.  Cf.  GOURSAT-HEDRICK,  Mathematical  Analysis,  Vol.  i,  p.  19.  —  S.  E.  R. 


§34-    CONFORMAL   REPRESENTATION  183 

the  equation 


€  becomes  infinitesimal  with  s»  —  z\  and  z3  —  z±.  If  e'  is 
omitted  in  this  equation,  we  obtain  (except  for  the  symbols) 
equation  (17)  of  §  10,  whose  geometrical  significance  was  dis 
cussed  there.  We  thus  have  an  answer  to  the  proposed  ques 
tion  ;  it  can  be  formulated  as  follows  : 

I.  If  w  is  a  regular  function  of  z,  then  every  triangle  of  the 
z-p!ane  whose  sides  are  infinitesimals  of  the  same  order,  is  similar  to 
the  corresponding  triangle  of  the  w-plane  up  to    infinitesimals  of 
higher  order,  that  is,  ratio  of  sides  and  angles  of  the  one  differ 
only  by  infinitesimals  from  the  corresponding  parts  of  the  other. 

In  particular,  if  we  apply  the  results  of  §  10  for  the  finite 
triangles  discussed  there  to  the  infinitesimal  triangles  just  men 
tioned,  it  follows  that  : 

II.  The  absolute  value  of  the  derivative  -  -  at  a  point  of  the 

dz 

z-plane  gives  the  scale  of  similarity*  at  that  poi  tit,  that  is,  gives  the 
factor  by  which  the  length  of  an  infinitesimal  arc  of  the  z-plane 
must  be  multiplied  in  order  to  obtain  the  length  of  the  corresponding 
arc  of  the  w-planc. 

III.  The  amplitude  at.  of  —  gives  the  angle  through  which  each 

dz 

element  of  arc  at  the  point  z  must  be  turned  in  order  to  be  made 
parallel  to  the  corresponding  element  of  arc  of  the  w-plane. 


Since  this  angle  depends  only  upon  the  point  s,  and  not  upon 
the  direction  of  the  element  of  arc,  it  follows  that 

IV.  Any  two  curves  of  the  z-plane  form  with  each  other  at  each 
of  their  points  of  intersection^  the  same  angle  as  the  cor  responding 
curves  of  the  w-plane  at  the  corresponding  points  of  intersection', 
*  Sometimes  called  the  cartographic  modulus,  —  S.  E.  R. 


1  84  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

or  (by  using  the  terminology  introduced  in  VII,  §  n)  : 

V.  A  domain  of  the  z-plane  in  which  a  regular  function  w  with 
a  complex  argument  z  is  defined,  is  mapped  conformally  by  means 
of  this  function  on  a  region  of  the  w-plane. 

We  have  already  become  acquainted  with  a  large  number  of 
such  representations  in  Chapter  II  ;  in  what  follows  we  shall  find 
many  others. 

The  inference  from  (i)  to  (2)  is  only  permissible  when  the 

limit  of  (  z<i  ~  Zl  \  is  finite,  and  hence  that  of  (w*  ~  W{\  is  dif- 


ferent  from  zero.  Since  we  have  to  do  here  only  with  triangles 
all  of  whose  sides  are  infinitesimals  of  the  same  order,  this 
inference  is  true  when  and  only  when 


is  different  from  zero.     Consequently,  the  following  corollary 
must  be  added  to  Theorem  II  : 

VI.    The  conformality  of  the  representation  is  not  established  at 
those  places  at  which  , 


The  relation  which  the  angle  at  such  a  point  in  the  one  plane 
bears  to  the  corresponding  angle  in  the  other  plane  will  be  dis 
cussed  in  §  69. 

In  many  cases  it  is  of  interest  to  notice  what  curves  of  the 
w-plane  correspond  to  the  parallels  to  the  coordinate  axes  of  the 
z-plane.  The  equations  of  these  curves  'are  obtained  if  we  put 
w=f(z)  =  <f>(x,  y)  +  i(j/(x,  y)  and  then  eliminate  x  and  y  respec 
tively  from  the  equations  : 

(3)  4>(x,y)  =  u,    $(x,y)  =  v, 


§34-    CONFORM AL   REPRESENTATION  185 

Conversely,  the  equations 

(4)  <f>(x,  })  =  const. 

and 

(5)  $(x,  y)  =  const. 

represent  those  systems  of  curves  of  the  s-plane,  to  which  the 
parallels  to  the  coordinate  axes  of  the  zt'-plane  correspond. 
Since  the  representation  is  conformal,  every  curve  of  the  system 
(4)  intersects  every  curve  of  system  (5)  at  right  angles.  The  two 
systems  of  curves  are  orthogonal  to  each  other. 

Further,  if  we  choose  from  the  systems  of  parallels  to  the 
coordinate  axes  in  the  w-plane  such  a  distinct  set,  the  lines  of 
which  are  at  the  same  constant  distance  from  each  other  in  both 
systems,  they  will  divide  the  w-plane  into  squares ;  these  cor 
respond  to  divisions  of  the  s-plane,  which  differ  less  and  less 
from  squares,  the  smaller  that  constant  distance  is  chosen.  This 
property  of  the  systems  (4)  and  (5)  is  usually  expressed  more 
briefly  by  saying :  They  divide  the  z-plane  into  indefinitely  small 
squares.  A  system  of  curves,  for  which  a  second  system  can 
be  found  such  that  the  two  together  divide  the  plane  (or  in  gen 
eral  any  surface)  into  indefinitely  small  squares,  is  called  an 
isometric  or  an  isothermal  system. 

This  latter  terminology  is  well  suited  to  the  physical  interpre 
tation  of  such  a  system  of  curves  which  we  must  at  least  mention. 
Let  us  suppose  a  (ponderable  or  imponderable)  fluid  flowing  in 
the  jrr-plane,  and  let  £,  rj  be  the  x-  and  ^components  of  its 
velocity  at  some  point  (x,  y).  Let  us  fix  in  mind  a  rectangle 
whose  sides  are  parallel  to  the  coordinate  axes  and  having  the 
distances  x,  x  +  dx,  y,  y  +  dy  respectively  from  them.  In  the 
time  dt,  the  mass  %dtdy  will  flow  in  over  the  side  (x),  and  during 
the  same  time  there  flows  out  over  the  opposite  side  the  mass  of 

liquid  (&  +  —  dx\dtdy.     Likewise   over  the  side   (j-),   the   mass 


1  86  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 


t]dtdx  comes  in  ;  over  the  opposite  side  (  rj  -f-  -^  dydtdx   goes 

V        dy      J 

out.     Therefore,  in  the  time  dt  the  mass  of  liquid  contained  in 
the  rectangle  dxdy  is  increased  by 


If  the  liquid  be  regarded  as  incompressible,  an  increase  or  a 
decrease  in  the  mass  of  the  liquid  contained  in  the  rectangle 
cannot  take  place  and  hence  it  must  follow  that 

(6)  F+F"  =  °- 

ox      ay 

And  if  £,  -rj  are  the  derivatives  of  one  and  the  same  function 
v(x,  y)  (the  "velocity  potential")*  with  respect  to  the  co 
ordinates,  that  is,  ^  _dy 

^~dx'    V-Ty* 
it  follows  that 

(7)  ^_^  =  o. 

dy      dx 

The  two  equations  (6)  and  (7)  together  tell  us  that  £  =  rj  +  /£  is 
a  function  of  the  complex  argument  z  =  (x  +  iy)  ;  and  that  iv  is 
the  imaginary  part  of  the  function  I  £dz  (defined  in  the  next 
section).  If  u  be  taken  as  the  real  part  of  this  function,  it 
follows  that 

/ON  t  du  du 

(8)  €  =  —  ^-  ,    rj  =  —  : 

dy  dx 

in  other  words,  the  direction  of  the  velocity  at  any  point  coin 
cides  with  the  tangent  to  the  curve  u  =  const,  going  through 
this  point.  These  curves  are  then  the  lines  of  flow.  We  thus 
find  that  : 

*  Cf.  HARKNESS  AND  MORLEY,  Introduction,  etc.  p.  315;  OSGOOD,  Lehrbuch 
der  Funktionentheotie,  Vol.  I,  chap.  13.  —  S.  E.  R. 


§34-    CONFORMAL    REPRESENTATION  1 87 

The  "  lines  of  level'1'1  (lines  of  equipotential}  v  =  const,  and  the 
"  lines  of  flow  "  u  =  const,  for  a  constant  current  of  an  incompressi 
ble  fluid  in  the  plane,  which  has  a  velocity  potential,  together  divide 
the  plant  into  indefinitely  small  squares. 

Conversely,  if  we  have  given  a  regular  function  w  =u  +  iv  of  the 
complex  variable  z  =  x  +  n1,  we  can  always  look  upon  the  curves 
u  =  const. ,  v  =  const,  as  lines  of flow>  and  lines  of  Iwel  for  a  con 
stant  non-rotating  current  of  an  incompressible  fluid  in  this  part  of 
the  plane. 

For  transmission  of  heat,  temperature  takes  the  place  of 
velocity  potential ;  for  the  transmission  of  electricity,  the  term 
electrical  potential  is  used. 

EXAMPLES 

1.  The  lines  of  flow  and  lines  of  level  are  sometimes  called 
path-curves  and  niveau  lines  respectively.  We  may  define  a 
path-curve  of  a  linear  transformation  to  be  any  curve  in  the 
plane  which  is  transformed  into  itself  by  the  transformation. 
This  does  not  imply  that  the  points  on  the  curve  remain  fixed. 

A  system  of  niveau  lines  is  a  set  of  lines  each  of  which  is 
transformed  into  the  next  of  the  set.  The  niveau  lines  are  usu 
ally  but  not  necessarily  chosen  so  as  to  meet  the  path-curves  at 
right  angles. 

For  the  transformation  z'  =  z  +  a,  the  path-curves  are  the 
lines  parallel  to  ~oa  (o  is  the  origin),  and  a  set  of  niveau  lines  is 
the  line  through  o  perpendicular  to  0a  and  the  lines  parallel  to 
it,  at  distances  |  oa  \  from  each  other. 

For  z'  =  az,  first  let  a  be  real.  The  path-curves  are  then  the 
lines  through  o  and  a  system  of  niveau  lines  is  the  set  of  circles 
with  o  as  center  and  radii  k,  ka,  ka^,  ka*,-~  kan,  where  k  has  any 
real  value.  Second,  let  !  a  =  i.  The  figure  in  the  preceding  case 
is  reversed,  path-curves  becoming  niveau  lines  and  vice  versa. 


1 88  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

Third,  let    a    =£  i  and  am  a  3=  o.     In  this  case  the  path-curves 
are  logarithmic  spirals  with  centers  at  o  whose  equations  are 

/,        am  a      , 

"  =  ioTM-logr+"' 

where  n  has  any  real  value.     The  corresponding  niveau  lines 
are  the  log  spirals 

/!      log:  I  a  \    i 
—  0  =  —     — •••  log  r  +  n. 
am  a 

Cf.  HARKNESS  AND  MORLEY,  Introduction,  pp.  55,  56. 

2.    What  curves  of  the  w-plane  correspond  by  the  transforma- 

r,      I        j 

tion  z0/ = to  the  lines  #  =  o,  ±i,  jy  =  o,  ±  i,  and  to  the 

unit  circle  of  the  z-plane  ? 

§  35.    The  Integral  of  a  Regular  Function  of  a  Complex 
Argument 

I.    The  integral  of  a  complex  functio?i  u  -\-  iv  with  respect  to  a 
real  variable  t  between  the  real  limits  a,  b 


(i) 

we  understand  to  be  (cf.  §  28) ; 

f  udt+i\    vdt. 

But  what  is  to  be  understood  by  an  integral  between  complex 
limits  requires  some  explanation.  A  real  variable  of  integra 
tion  can  pass  from  its  lower  to  its  upper  limit  (through  one  se 
quence  of  intermediate  values)  along  only  one  path  (providing 
the  path  does  not  pass  through  infinity  and  that  it  is  not  re 
traced  anywhere  along  it).  On  the  contrary,  we  can  pass  from 


§35-    THE  INTEGRAL  OF  A   REGULAR   FUNCTION       189 

one  value  of  a  complex  variable  to  another  through  many  differ 
ent  sequences  of  intermediate  values  ;  we  can  connect  two 
points  of  the  plane,  upon  which  they  are  represented  geometri 
cally,  by  many  different  curves.  Therefore,  in  speaking  of  an 
integral  between  complex  limits,  we  must  necessarily  fix  upon 
a  path  of  integration  and  regard  the  integral  as  a  curvilinear 
integral  of  the  kind  defined  in  §  29.  Accordingly, 

II.  If  there  is  given  a  path  T  connecting  the  points  ZQ  =  x0  -f  /Y0 
and  zl  =  Xi  +  y'i»  and  if  upon  this  path  iv  =  u  +  iv  is  a  continuous 
complex  function  of  x  and  v,  then  we  understand 


(2) 

to  be  the  integral 

f  0  -f  iv)(dx  +  idy)  =  f  (*&  -  vdy)  +  /  C(vdx  +  udy). 
Jr  JT  «/r 

The  question  frequently  arises  whether  there  is  an  upper 
limit  to  the  absolute  value  of  a  complex  integral.  In  this  con 
nection  Theorem  IV  of  §  5  will  aid  us  ;  it  follows  from  it  that 


in  which  j  dz  \  is  the  element  of  arc  of  the  path  of  integration; 
the  right-hand  side  is  therefore   ^  ^ 

where  M  is  the  maximum  of  w  on  the  path  of  integration  and 
L  the  length  of  this  path. 

For  example,  between  the  limits  ZQ  and  z  (cf.  VI,  §  29), 

J  dz  =  \  dx  +  i(  dy  =  X  —  XQ  +  i(y  —  _r0), 
Czifz  =  C(xdx  -  ydy)  -f  /  f(xdy  +  ydx) 


I QO  IV     SINGLE- VALUED   ANALYTIC   FUNCTIONS 

for  any  arbitrary  path  of  integration.  These  two  integrals  are 
thus  independent  of  the  path.  As  a  direct  consequence  of  VI, 
§  29,  the  following  general  theorem  holds: 

III.    If  f(z)  is  the  derivative  of  a  function  F(z]  of  z  regular  in 
a  simply  connected  domain  B,  then 


however  the  path  from  ZQ  to  zl  inside  this  domain  may  be  chosen. 

Further,  the  following  theorem  holds  : 

IV.    If  a  function  f(z)  of  a  complex  argument  is  regular  in  a 
simply  connected  domain  B,  then 


(4) 

for  every  closed  curve  which  lies  entirely  inside  of  B 
For,  according  to  hypothesis, 


for  every  point  z0  of  the  domain  ;  a  neighborhood  about  every 
such  point  can  then  be  so  chosen  that 

<  £ 


for  all  points  ZQ  +  £  of  this  neighborhood  ;  hence  if  we  put 


t]  is  a  function  of  z  and  £  respectively,  whose  absolute  value  is 
smaller  than  e  for  all  points  of  the  neighborhood  of  z.     Inte- 

*  Many  proofs  have  been  given  of  this  fundamental  theorem  in  the  theory  of 
functions.  The  reader  will  be  interested  in  comparing  the  proof  given  here  with 
the  one  due  to  GOURSAT,  Acta  Math.,  Vol.  IV,  p.  197.  —  S.  E.  R. 


§  35-    THE   INTEGRAL   OF  A   REGULAR   FUNCTION       IQI 

grating  now  about  a  square  whose  length  of  side  is  8  and  which 
belongs  entirely  to  this  neighborhood,  introducing  £  as  variable 
of  integration,  we  obtain  : 


The  first  two  integrals  on  the  right-hand  side  are  equal  to  zero 
and  the  last  is  in  absolute  value  less  than  8  •  A/2  •  e  •  4  8,  that  is, 

<  4  V2  82e. 

But  from  this  according  to  IX,  §  29,  it  follows  that  the  integral 
taken  over  any  closed  curve  lying  entirely  inside  of  B  is  zero. 

Q.E.D. 

If,  therefore,  we  have  two  paths  ABC  and  ADC  inside  of  this 
domain  B  and  between  the  same  two  points  A  and  C(cf.  Fig.  15), 
it  follows  that 

f  /(z\/z  +  f  f(zyz  =  a 

JABC  JCDA 

or  (by  II,  §  29)  :      f  f(z]dz  =  f  f(z)dz, 
•.'ABC  J  ADC 

that  is,  the  following  form  of  Theorem  III  is  also  true  : 

V.  If  we  consider  only  suth  paths  of  integration  which  lie  entirely 
within  a  simply  connected  domain  B  in  which  the  function  f(z)  is 
regular,  then  the  value  of  the  integral 

(5) 

is  independent  of  the  path,  dependent  only  upon  the  initial-  and  end- 
points  ZQ  and  ZK 

If  we  keep  the  end-point  fixed,  we  can  regard  the  value  of  the 
integral  in  the  sense  of  definition  I,  §  31,  as  a  complex  function 
of  the  upper  limit  and  as  such  designate  it  by  F(sl).  To  obtain 
then  the  value  of  this  function  for  a  neighboring  argument  %+£ 


I  Q2  IV.    SINGLE-  VALUED    ANALYTIC   FUNCTIONS 

(which  also  belongs  to  this  domain),  we  can  take  as  the  path  of 
integration  from  z0  to  z1  +  £  any  suitable  path  from  z0  through  % 
since  the  value  of  the  integral  is  independent  of  the  path  ;  we 
thus  obtain  : 


4- 


f  %V*  = 


Since  f(z)  is  by  hypothesis  continuous,  £  can  be  taken  so  small 
that 

!/(*)-  /(%)!<« 

for  all  points  of  the  path  from  z±  to  zl  +  £  ;  then,  according  to  (3)  : 

(6)  |^l  +  0-^,)-f/(2i)[<€      £ 

and,  therefore,  in  whatever  manner  £  converges  to  zero, 

(7)  ' 


that  is,  according  to  I,  §  33  : 

VI.  Under  the  hypotheses  of  Theorem  V,  the  value  of  an  integral 
is  a  regular  function  of  its  upper  limit  :  and  its  derivative  is  the 
function  to  be  integrated. 

We  add  further  the  corollary  : 

VII.  If  two  curves  F,  y  inclose  an  annular  domain  B  in  which 
the  function  f(z)  satisfies  the  conditions  of  Theorem  III,  then 

(8) 

provided  we  pass  along  the  curves  so  that  the  area  inclosed  by  each 
of  them  always  lies  to  the  left. 


§  35-    THE   INTEGRAL  OF  A    REGULAR   FUNCTION       193 


To  prove  this  theo 
rem  let  us  think  of  the 
domain  B  cut  along  a 
line  C  which  connects 
a  point  a  of  y  with  a 
point  A  of  T.  By  this 
means  we  obtain  a 
simply  connected  do 
main  B.  To  pass  now 
around  these  bounda 
ries  keeping  the  do 
main  B  always  to  our 
left,  we  proceed  as  follows  along 

1.  The  curve  T  in  the  direction  of  the  arrow  ; 

2.  The  curve  Cfrom  A  to  a\ 

3.  The  curve  y  opposite  to  the  direction  of  the  arrow; 

4.  The  curve  C  from  a  to  A. 

The  sum  of  the  integrals  taken  along  these  four  curves  is, 
according  to  Theorem  V,  equal  to  zero.  But  since  the  second 
of  these  four  integrals  is  equal  but  opposite  in  sign  to  the  fourth, 
it  follows  that  : 


FIG.  16 


when  the  integral  is  taken  along  the  two  curves  as  above  indi 
cated.  But  in  Theorem  VII  the  direction  on  y  was  opposite  to 
this,  on  account  of  which  we  must  there  use  the  opposite  sign. 

As  an  example  of  the  methods  of  this  paragraph,  let  us  treat 
the  problem  to  determine  the  value  of  the  integral  : 


taken  along  any  curve  T  inclosing  the  point  £  =  £  +  /iy,  when  n  is 
a  positive  or  negative  integer.     Let  us  draw  about  £  a  circle  C 


254  IV-    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

which  it  is  regular;  hence  expand  for  a  circle  with  center  at  o,  that  is,  in 
powers  of  z.  Substitute  these  in  f(z)  and  the  expansion  for  the  domain  B 
is  obtained. 

11.  Suppose  f(z]  and  (j>(z)  have  at  the  point  z  =  a  poles  of 
order  m  and  n  respectively.     What  can  be  said  of  the  behavior 

of  the  functions  fi7\ 

/(*)  •  *(*),  /(*)  +  *(*),  "^ 

£(*) 

at  this  point  ?     Discuss  all  cases. 

12.  Suppose  /(z)  has  an  w-fold  zero  at  z  —  a.     Show  that  the 
integral 


has  an  (w  +  i)-fold  zero  there. 

State  the  analogous  proposition  for  the  integral 


in  the  neighborhood  of  a  pole  a. 

§  48.   Behavior  of  a  Regular  Function  in  the  Neighborhood  of 
a  Critical  Point 

We  may  frequently  prove  that  a  function  is  in  general  regular  in 
a  domain,  but  the  proof  may  fail  for  particular  points  of  this 
domain,  so  that  the  question  as  to  the  behavior  of  the  function 
at  these  critical  points  remains  undetermined.  A  certain 
amount  of  information  is  furnished  in  such  cases  by  the 
LAURENT'S  series. 

Let  the  origin  be  such  a  point,  that  is,  let  the  function  f(z)  to 
be  investigated  be  regular  at  every  point  of  a  certain  neighbor 
hood  of  the  origin  with  the  exception  of  the  origin  itself,  con 
cerning  which  nothing  is  known.  The  circle  y  used  in  connec 
tion  with  LAURENT'S  theorem  can  then  be  taken  arbitrarily 
small. 


§48.    A   REGULAR   FUNCTION  NEAR   A  CRITICAL   POINT    255 

And  when  |  f(z]  \  always  remains  less  than  an  assignable  limit 
however  near  z  may  approach  the  origin,  it  follows  that  the  coeffi 
cients  a_n  (5,  §  47)  must  all  be  equal  to  zero.  But  then  the 
LAURENT'S  expansion  of  f(z)  represents  a  function  regular  at 
the  origin ;  and  if  removable  discontinuities  be  excluded  as 
agreed  upon  in  §  43,  it  follows  that  this  function  must  coincide 
with/(s)  even  at  the  origin.  Hence  the  following  theorem  : 

I.  When  a  function  of  a  complex  argument  is  regular  in  the 
neighborJwod  of  the  origin,  this  point  itself  excepted,  and  when, 
in  arbitrarily  approaching  the  origin,  it  remains  in  absolute 
value  always  less  tJian  any  assignable  limit,  then  the  function  is 
regular  at  the  origin  itself  provided  that  removable  discontinuities 
are  excluded. 

This  may  be  expressed  more  briefly  but  less  exactly  as  follows  : 
"  A  function  of  a  complex  argument  is  everywhere  continuous 
where  it  is  finite." 

But  if  in  the  LAURENT'S  expansion  of  the  function  in  the 
neighborhood  of  the  point  z  =  o  terms  with  negative  exponents 
appear,  we  must  determine  whether  there  are  an  infinite  or  only 
a  finite  number  of  such  terms.  In  the  first  case  the  function 
behaves  at  the  point  z  =  o  just  as  a  transcendental  integral 
function  at  infinity  (X,  §  44)  ;  that  is,  it  approaches  arbitrarily 
near  to  every  value  in  every  neighborhood  of  this  point.  For, 
the  sum  of  the  terms  with  positive  exponents  becomes  arbitrarily 
small  in  a  sufficiently  small  neighborhood  of  the  point  z=o  and 
it  is  only  a  question  of  the  terms  with  negative  exponents.  In 
the  second  case  the  function  is  definitely  infinite  at  z  =  o  in  the 
following  sense : 

When  a  positive  number  M  however  large  is  given,  we  can  always 
draw  a  circle  about  the  point  2  =  0  with  a  radius  sufficiently  small 
(but  >  o)  so  that\f(z)  \  >  M  for  all  points  inside  of  it.  But,  if  in 


196  IV.    SINGLE- VALUED   ANALYTIC   FUNCTIONS 

and  if  in  the  second  integral  we  put  likewise : 
(2)  z  —  £  ==  r(cos  /+  /  sin  /), 

dz  =  /r(cos  t-\-i  sin  t)dt> 

dz        ,, 

—  =  utt, 


we  find  ((3),  §  35)  that  its  absolute  value  is 

r2jr 

<  e  I      dt,  that  is,  <  2  TTC, 
~   »/o 

that  is,  it  can  be  made  smaller  than  any  arbitrary,  previously 
assigned  quantity  by  taking  r  sufficiently  small.  But  the  value 
of  the  left  side  of  equation  (i),  as  also  2  TT//(£),  is  independent 
of  r\  if  the  difference  of  these  two  quantities  were  different 
from  zero,  it  could  not  be  made  smaller  than  any  limit  by 
making  r  smaller.  It  follows  accordingly  that 


(3) 


I.  By  means  of  this  formula  stated  by  CAUCHY,  the  value 
which  a  regular  function  of  a  complex  argument  z  has  at  any  point 
£  of  a  domain  B,  is  expressed  by  the  value  of  the  same  function  on 
the  bounding  curve  F  of  the  domain. 

The  conclusion  from  this  theorem  is  not  that  we  can  assign 
arbitrarily  the  values  of  such  a  f  unction  f(z)  on  the  boundary: 
of  course  formula  (3)  would  then  always  furnish  a  f  unction  /(£) 
regular  in  the  interior  of  the  domain,  but  this  function  would 
not  in  general  converge  to  the  value  preassigned  at  a  point  on 
the  boundary  as  £  approaches  this  point. 

If  the  curve  F  is  a  circle  whose  center  is  £,  and  if  we  put 
f(z)  =  u  +  iv  and  /(£)  =  UQ+  iv^  introduce  substitution  (2)  in 


§  36.    CAUCHY'S  THEOREM  197 

equation  (3)   and  eqliate  real  and    imaginary  parts,  it   follows 

that:  r,  r, 

//0  =  —      j      udt  and  r0  =  —      I      vdt. 

2   7T*/0  2   7T«^0 

The  first  one  of  these  equations  expresses  the  fact  that : 

II.  The  value  of  the  real  part  of  a  regular  function  of  a  complex 
argument  at  the  center  of  a  circle,  is  equal  to  the  mean  of  its  values 
taken  along  the  circumference. 

Thus  it  can  neither  be  greater  than  all  these  values  nor  less 
than  all  of  them.  It  therefore  follows,  provided  the  radius  of 
the  circle  is  taken  sufficiently  small,  that : 

III.  The  real  part  of  a  function  of  a  complex  argument  regular 
in  a  domain  B,  can  never  have  a  maximum  nor  a  minimum  at  an 
inner  point  of  this  domain. 

The  same  theorems  hold  of  course  for  v. 

EXAMPLES 
1.    Evaluate  the  integral 


f, 


r 

extended  around  any  closed   curve  in    the  z-plane  which  does 
not  pass  through  the  point  s  =  o. 

2.    Compute    I  zdz  where  P  is  a  straight  line  from  z  =  o  to 
Jp 


z  =  a  +  >. 


HINT.  —  (  zdz  =  \   O  +  (j»)  (dx  +  idy)  ;   express  jp  and  dy  in  terms  of  x  and 
dx  and  take  the  limits  on  the  integration  from  o  to  a. 

3.    Find  the  value  of   I  zdz  where  C  is  a  circle  whose  center 
Jc 

is  at  the  origin  and  whose  radius  =  i. 


250  IV.    SINGLE-VALUED   ANALYTIC  FUNCTIONS 

Conversely,  let  us  suppose  that  for  a  function  /(£)  a  develop 
ment  of  the  form  (3)  is  found  which  converges  inside  of  the 
annular  domain  between  two  circles  T,  y;  and  in  fact,  to  fix 
this  hypothesis  more  precisely,  let  each  of  the  two  series 


be  convergent  inside  of  the  annular  domain.  Then  the  first 
series,  according  to  III,  §  38,  converges  uniformly  in  every 
domain  which  lies  entirely  inside  of  T,  the  second  converges 
uniformly  in  every  domain  which  lies  entirely  outside  of  y. 
Hence  both  series  converge  uniformly  on  a  curve  such  as  C  in 
Fig.  23,  and  hence  they  may  be  integrated  term  by  term  along 
this  curve.  Let  us  do  this  after  first  multiplying  by  £-m~l ;  then, 
in  connection  with  equations  (10)  and  (n)  of  §35,  we  find: 

(9) 

and  this  coincides  with  (7)  ;  that  is,  therefore, 

II.  When  a  function  can  be  developed  in  a  series  of  the  form  (j) 
which  converges  in  the  given  sense  inside  of  the  circular  ring  be 
tween  T  and  y,  then  the  coefficients  have  the  values  given  by  (/)  ; 
this  development  is  therefore  unique. 

The  last  statement  requires  some  explanation  in  order  that  it 
may  have  only  the  intended  meaning.  A  function  may  be 
regular  inside  of  different  circular  rings,  e.g.,  between  yl  and  y2 
between  y2  and  y3,  while  upon  y2  there  are,  for  example,  poles  of 
the  function.  Theorem  I  is  then  applicable  to  each  of  these  two 
rings  and  two  LAURENT'S  expansions  are  thus  obtained,  one  of 
which  converges  between  yl  and  y2  and  the  other  between  y2 
and  y3 ;  and  we  are,  therefore,  not  to  understand  Theorem  II  to 


§47-    THE   LAURENT'S   SERIES  2$ I 

mean  that  these  two  expansions  must  have  the  same  coefficients. 
On  the  contrary.  Theorem  II  is  applicable  only  to  the  expansion 
inside  of  one  and  the  same  ring. 

Thus,  for  example,  we  obtain  for  the  expansion  of 


Z*  —  32+2        Z  —  2        2—1 

inside  of  the  circle  of  unit  radius  about  the  point  z  =  o  : 


between  this  circle  and  the  circle  of  radius  2  : 


outside  of  the  latter  :      +  -L  +  1  +  1  -|-  .  .  .  . 

z*     2s     24 

The  generalization  of  the  theorems  of  this  paragraph  to  the 
case  where  the  two  concentric  circles  have  not  the  point  z  =  o 
but  any  other  arbitrary  point  as  center  is  treated  as  in  VI,  §  39, 
and  requires  no  further  explanation. 

EXAMPLES 

1.  Develop  -  -  in  a  series  of  integral  powers  of  z 

2-3      *-  J 
valid  for  the  domain  in  which  this  function  is  regular. 

2.  Expand  -     -  inside  a  circle  whose  center  is  O  ;  that  is, 

i  —  z 

expand  in  powers  of  z.     How  large  may  the  circle  of  conver 
gence  be  ? 

3.  Expand  -  inside  a  circle  whose  center  is  the  point  /  ;  that 

z 

is,  in  powers  of  z  —  i. 


2OO 


IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 


will  be  satisfied  by  the  same  value  of  n  (and  all  larger  values) 
for  the  same  e  and  for  every  value  of  z  whose  absolute  value  is 
<^  r ;  for,  |  zn  ^  rn  and  |  i  —  z  \  >  i  —  r  for  all  such  values. 
We  can  then  state  the  following  theorem  on  the  basis  of  the 
definition  of  uniform  convergence  (A.  A.  §  66) : 

II.    The  series  (i)  converges  uniformly  in  every  circle  about  the 
origin  with  radius  <  i. 

After  proving  this  introductory  the 
orem,  we  return  now  to  equation  (3), 
§  36.  Let  us  suppose  for  the  sake  of 
simplicity  that  the  origin  lies  on  the 
inside  of  the  domain  defining  the 
function  f(z] ;  we  can  then  choose  for 
the  curve  F  a  circle  about  the  origin 
with  a  sufficiently  small  radius.  Then 

\t  <\* 

for  all  points  £  within  this  circle  and  for  all  points  z  upon  it ; 
accordingly,  by  I  and  II,  the  series 

i+i+e+...+ii+... 

„  yl  iy&  r^  +  1 

xy  .6  &  /y 


FIG.  18. 


converges  uniformly  to  — 

for  all  these  values  of  £  and  z.  Moreover,  the  uniformity  of  the 
convergence  holds  if  we  multiply  all  the  terms  by  f(z)-  There 
fore,  by  VIII,  §  28,  we  may  integrate  the  series  thus  formed, 
term  by  term,  along  the  circumference  of  the  circle.  We  obtain 
accordingly : 

(4)  2 


§  38.    PROPERTIES  OF   COMPLEX   POWER   SERIES       2OI 

and  along  with  this  the  theorem  : 

III.  If  a  function  of  a  complex  argument  is  regular  in  a  circle 
about  the  origin,  it  can  then  be  developed,  for  all  points  £  WITHIN 
this  circle,  in  a  convergent  series  of  powers  of  £  with  positive,  inte 
gral,  increasing  exponents. 

The  theorem,  however,  says  nothing  about  the  behavior  of 
the  series  upon  the  circumference  of  the  circle. 

In  evaluating  the  integrals  in  (4),  the  circle  T  can  be  replaced, 
according  to  VII,  §  35,  by  any  other  curve  about  the  origin 
provided  that  inside  of  this  curve  the  function  f(z)  is  regular. 

§  38.   Properties  of  Complex  Power  Series 

In  connection  with  the  results  of  the  previous  paragraph  the 
converse  question  arises,  whether  a  "  Power  Series  "  of  the  kind 
considered  there  always  represents  a  regular  function  of  the 
argument.  Let  such  a  series  be  represented  by 

cc 

(i)  2Xs"  =  aQ  +  a&  +  a.z1  +  -  •  -  +  anzn  +  •  •  •  ; 

n=0 

we  inquire  first  about  its  convergence.  It  converges  of  course 
for  z  =  o  ;  if  it  converges  for  no  other  value,  it  could  not  be 
used  as  the  definition  of  a  function. 

It  is  quite  possible  for  a  power  series  to  converge  "perma 
nently  "/  that  is,  to  converge  for  all  finite  values  of  z  (examples  of 
which  will  be  found  in  §  40).  It  then  represents  a  function 
regular  over  the  whole  plane  ;  conversely,  every  function  regular 
over  the  whole  plane  may  be  represented  by  such  a  permanently 
converging  power  series. 

According  to  WEIERSTRASS,  such  a  function  is  called  a 
transcendental  integral  function. 

If  the  series  converges  for  any  value  z  =  c  different  from  zero, 


246  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

taken  along  the  boundary  of  a  domain  in  the  z/^-plane,  represents, 
as  will  be  taken  for  granted  here,  the  area  of  this  domain ;  and 
it  has  the  positive  or  the  negative  sign  according  as  the  bound 
ary  is  described  in  the  positive  or  in  the  negative  sense  in  the 
process  of  integration.  If  we  introduce  x  and  y  as  variables  of 
integration  in  this  integral,  regarding  u  and  v  as  functions  of  x 
and  yt  we  obtain  the  integral : 


taken  along  the  corresponding  curve  of  the  jry-plane.  If  this 
curve  incloses  a  domain  whose  map  upon  the  corresponding 
domain  of  the  w^-plane  is  reversibly  unique,  then  the  value  of 
the  integral  is  positive  when  taken  around  the  domain  of  the 
^•j-plane  in  the  positive  sense,  and  negative  in  the  opposite  case 
if  the  sense  of  the  angle  remains  unchanged  throughout  the 
mapping.  The  first  is  always  the  case  according  to  the  last 
theorems  if  u  -f-  iv  is  an  analytic  function  of  x  +  iy  and  the 
domain  is  sufficiently  small.  But  since  an  integral  taken  over 
an  arbitrary  curve  can  always  be  replaced  as  in  §  29  by  a  sum 
of  integrals  over  sufficiently  small  curves,  it  follows  that : 

XIII.    If  u  -+-  iv  is  a  regular  function  of  x  -f-  iy  over  the  whole 
domain  inclosed  by  a  curve  F,  then  the  integral 


taken  in  the  positive  sense  along  F,  is  always  positive. 

The  only  exception  to  this  theorem  occurs  when  the  function 
u  +  iv  maps  the  domain  under  consideration  in  the  #+*y-plane 
not  in  general  upon  a  domain,  but  upon  a  single  point,  that  is, 
when  it  is  constant.  (The  conceivable  case  of  mapping  the 
domain  of  the  x  -f  /y plane  upon  a  curve  of  the  u  +  zV-plane  is 


§  47-    THE   LAURENT'S   SERIES  247 

not  possible  on  account  of  the  Theorems  V,  §  26  ;  VIII,  §  38  ; 
X,  §  46.)  To  include  this  exception  in  the  formulation  of 
Theorem  XIII,  we  must  say  "  never  negative  and  only  zero 
when  //  +  iv  is  constant  "  instead  of  "  positive." 

By  means  of  this  theorem  we  may  obtain  a  second  proof  of 
the  fundamental  Theorem  IV,  §  44.  Theorem  XIII  is  also 
valid  for  a  part  of  the  sphere  which  includes  the  point  oo  as  an 
inner  point,  provided  that  the  function  u  +  iv  is  regular  in  this 
domain  in  the  sense  of  definition  I  of  §  44.  However,  we  must 
in  this  case  take  for  positive  direction  of  integration  that  one 
for  which  the  domain  under  consideration,  as  also  the  point  at 
infinity,  lies  to  the  left. 

If  now  we  have  a  function  which  is  regular  over  the  whole 
sphere,  we  can  divide  the  sphere  into  two  parts  by  any  curve 
which  does  not  go  through  the  point  infinity,  and  we  can  then 
apply  Theorem  XIII  to  each  of  these  two  parts.  It  then  fol 
lows  first,  that  the  integral  cannot  be  negative  when  we  take 
the  part  lying  on  the  finite  part  of  the  sphere  always  to  the  left ; 
and  second,  that  it  cannot  be  negative  when  the  part  containing 
infinity  lies  to  the  left.  These  two  conditions  are  together  pos 
sible  only  when  the  integral  is  zero.  But  then  the  function 
u  +  iv  is  constant,  Q.  E.  D. 

§  47.     The  LAURENT'S  Series 

In  §  36  we  studied  CAUCHY'S  theorem  for  a  domain  6"  which 
had  one  bounding  curve.  We  return  now  to  this  theorem,  study 
ing  it  for  a  domain  S  in  which  the  function  f(z)  is  known  to  be 
regular  and  which  has  two  bounding  curves  F,  y  (cf.  Fig.  16). 
Equation  (3)  of  §  36  also  holds  in  this  case  ;  but  the  integration 
is  performed  along  each  of  the  curves  F,  y  in  such  direction 
that  the  domain  S  lies  to  the  left.  To  evaluate  this  integral  in 
the  positive  sense  along  each  of  the  two  curves,  we  must  change 


2O4  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

This  theorem  is  the  converse  of  CAUCHY'S  Theorem  III,  §  37. 

Since  we  represented  the  derivative  of  the  above  function  by 
a  power  series,  we  can  apply  the  same  methods  to  it  and  in  this 
way  prove  the  existence  of  a  second  derivative,  etc.  We  there 
fore  state  the  following  general  theorem  : 

VI.  Every  power  series  has  a?i  unlimited  number  of  successive 
derivatives  continuous  inside  of  its  circle  of  convergence. 

This  result,  in  connection  with  CAUCHY'S  theorem,  enables 
us  to  state  the  following  fundamental  theorem : 

VII.  Every  function  of  a  complex  argument,  which  is  regular  in 
a  given  domain,  has  an  unlimited  number  of  successive  derivatives 
continuous  within  this  domain;  and  these  derivatives  are  regular 
functions. 

If  instead  of  the  expression  "  regular  function  of  a  complex 
argument,"  we  use  only  its  meaning  in  terms  of  real  functions, 
this  theorem  is  stated  as  follows : 

Vila.  If  u  and  v  are  two  real  functions  of  x,  y  which  are  con 
tinuous  in  a  given  domain  and  which  have  continuous  first  deriva 
tives  satisfying  the  differential  equations 

du  _  dv    dv  _       du 
dx      dy    dx  By 

it  follows  at  once  from  this  that  they  have  an  unlimited  number  of 
successive  derivatives  continuous  within  this  domain. 

With  the  aid  of  these  results  the  theorems  deduced  in  §  34 
may  be  supplemented  at  important  places.  If  w  =  u  -f  iv  is  a 
regular  function  of  z  =  x  -f-  iy,  then  the  functional  determinant 


du     dv      dv     du  _  fdit\?  .   (dv  , 
!Tx  '  dy  ~  Ihc  "dy  ~  \dx)       \dx 


dw 
~dz 


§  38.    PROPERTIES   OF   COMPLEX   POWER   SERIES       2O5 

and  hence  is  not  negative.  Since  we  have  proved  (VII)  the 
continuity  of  the  derivatives  which  appear,  we  can  apply 
Theorem  IV  of  §  27  and  conclude  that: 

VIII.  If  w  =f(z)  is  a  function  which  is  regular  and  single- 
valued  in   a   domain    B   and  which    has   a    derivative   different 
from   zero  everywliere   in   this    domain,  then    the    values   which 
w  takes  on  in  B  cover  once  without  gaps  a  definite  region  C  of  the 
w-plane. 

Since  the  value  of  the  limit  —  is  the  reciprocal  of  the  value 

dw 

of  the  limit  —  ,  it  follows  further  that: 
dz 

IX.  z  is  also  a  function  of  w  regular  within  the  region  C. 

Besides  : 

X.  If  the  function  w  =/(z)  satisfies  the  provisions  of  TJuorem 
VIII  and  if  IV  =  <t>(w)  is  a  function  of  w  regular  in   C,  then 

is  also  a  function  of  z  regular  within  B. 


For,  from  the  existence  of  the  limits  —   and  -   we  infer 

dz  dw 

T  -ijr 

the  existence  of  the  limit  '  -  as  with  functions  of  real  variables. 

dz 

Finally,  the   following   theorems    are    proved    just  as  if   the 
variables  were  restricted  to  real  values  (A.  A.  I,  II,  §  77)  : 

XI.  If  a  given  power  series  converges  for  other  values  in  addi 
tion  to  z  =  O,  tJien  a  limit  p  can  be  so  chosen  for  the  absolute  value 
of  z  that  for  all  \z\  <  />,  the  first  term  of  the  series  whose  coefficient 
is  not  zero  is  greater  in  absolute  value  than  the  sum  of  all  tJie  re 
maining  terms. 

XII.  For  every  function  f(z]  regular  in  the  neighborhood  of  the 
point  z  =  o,  a  circle  can  be  drawn  about  z  =  o  with  a  radius  so 
small  that  no  zero  off(z)  lies  in  it,  except  possibly  z  =  o  itself. 


242  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 


By  applying  Theorem  III,  §  45,  to   —       we  obtain  the  follow 
ing  theorem  : 


IV.    The  integral          —  [£-* 
2  rriJ   fu 


/(*) 

taken  in  the  positive  sense  along  the  boundary  of  a  domain  in  which 
the  function  f(z]  is  everywhere  regular  except  at  poles,  is  equal  to 
the  number  of  zeros  of  f  (z]  in  this  domain  diminished  by  the  num 
ber  of  poles ;  every  zero  and  every  pole  is  to  be  counted  here  as  often 
as  its  order  of  multiplicity  indicates. 

Further,  we  find  from  Theorem  VI  of  §  45  that : 

V.  Every  rational  function  becomes  zero  as  often  as  infinite  upon 
the  sphere  (which  is  only  another  formulation  of  Theorem  III, 
§21); 

and  if  we  apply  it  tof(z)—c  instead  off(z),  we  find  that : 

VI.  A  rational  function  takes  on  any  arbitrary  value  c  just  as 
often  as  it  becomes  infinite. 

In  these  theorems  too,  multiple  zeros  or  poles  are  to  be 
counted  according  to  their  order  of  multiplicity  ;  the  expression 
"f(z)  takes  on  the  value  f(z)  =  c  n  times  at  the  point  z  =  a"  means 
that  c  is  the  first  term  in  the  development  of  f(z)  in  powers  of 
z—a,  for  which  terms  with  i,  2,  •  • .,  (n—  i)st  powers  of  (z  —  a) 
do  not  appear,  but  the  term  (z  —  d]n  is  present. 

In  particular,  a  rational  integral  function  of  the  nth  degree  is 
everywhere  regular  except  at  infinity  and  has  an  //-fold  pole  at 
infinity  ;  it  therefore  follows  from  Theorem  V  that : 

VII.  Every  rational  integral  function  of  the  nth  degree  has  n 
zeros ;  or,  expressed  otherwise  : 

VIII.  Every  algebraic  equation  of  the  nth  degree  has  n  roots. 
We  thus  have  a  second  proof  of  the  fundamental  theorem  of 

algebra  (cf.  VII,  §  44). 


§  46.  THE  NUMBER  OF  ZEROS  AND  OF  POLES    243 

It  follows  further  from  this  that  a  rational  fractional  function 
has  as  many  poles  as  its  degree  indicates  (II,  §  20).  For,  if  the 
degree  m  of  the  numerator  is  not  greater  than  the  degree  ;/  of 
the  denominator,  its  degree  is  n  ;  it  is  then  regular  at  infinity 
and  has  ;/  poles  in  the  finite  part  of  the  plane.  But  if  m  >  n, 
its  degree  is  equal  to  m  and  it  has  an  (m  —  «)-fold  pole  at  infin 
ity  in  addition  to  the  ;/  poles  in  the  finite  part  of  the  plane. 
From  Theorem  VI  it  thus  follows  that  : 

IX.  Every  rational  function  takes  on  any  arbitrary  complex 
value  as  of  fen  as  its  degree  indicates. 

We  make  further  use  of  Theorem  IV  in  order  to  deduce  an 
important  extension  of  Theorem  VIII  of  §  38.  Let  w  =/(z)  be 
a  function  regular  in  a  circle  about  the  origin  and  f  (o)  =j=  o  ; 
without  loss  of  generality,  we  may  assume  that  w  =  o  for  z  =  o, 
since  this  can  always  be  obtained  by  a  parallel  translation  of 
the  7oplane.  We  can  then  take  /-  so  small,  according  to  VIII, 
§  39,  that  no  other  zeros  of  /(z)  lie  inside  or  upon  the  circum 
ference  of  a  circle  T  of  radius  r,  and  thus,  according  to  IV  : 


If  therefore  m  be  the  smallest  value  which  |  f  (z)  \  assumes  on  T, 
and  Wi  any  value  of  w  whose  absolute  value  is  smaller  than  m1 
then  the  number  of  roots  which  the  equation 

f(z)  =  u>, 
has  inside  of  F  is  : 


(4)  *=—  '  dz' 

2  TTlJr  f(z)  —  1L\ 

If  we  put 

(5)  --*W-* 


208 


IV.    SINGLE-  VALUED   ANALYTIC   FUNCTIONS 


Moreover,  we  obtain  from  4,  §  37,  on  the  basis  of  Theorem  II, 
the  following 

IV.    Expressions  for  the  value  of  the  function  and  its  derivatives 
at  the  origin  in  the  form  of  definite  integrals : 


(3) 


f 


2   7T 


]dz 


-»  +  l 


All  these  integrals  are  to  be  taken  along  a  circle  which  belongs 
entirely  to  the  interior  of  the  domain  in  which  the  function  is 
regular,  and  which  surrounds  the  origin  once  ;  according  to  VII, 
§  35,  they  can  be  taken  along  any  other  curve  of  the  domain 
inclosing  the  origin  instead  of  this  circle. 

From  Theorem  I  and  the  representation  by  integrals  in  (3) 
we  may  obtain  inequalities  for  the  coefficients  of  a  power  series 
which  we  shall  need  later,  and  on  this  account  we  deduce  them 
at  this  point.  If  M  is  the  upper  limit  of  the  absolute  values 
which  a  function  takes  on,  on  a  circle  of  radius  r  and  on  which 
the  series  is  convergent,  it  then  follows  that : 


dz 


Mr-' 


2  7T 


But 


dz 


—  d$,  if  we  put  z  —  r(cos  <f>  +  i  sin  <£)  (cf.  8,  9,  §  35); 


and  therefore: 

(4) 


an  |  <  Mr~n. 


§  39-  TAYLOR'S,  MACLAURIN'S  COMPLEX  POWER  SERIES     2OQ 

We  thus  obtain  the  following  theorem  : 

V.  If  r  and  M  have  the  meaning  given  them  above  for  a  pou>er 
series,  then  the  coefficients  of  this  power  series  satisfy  inequality  (4)  ; 
in  other  words,  their  absolute  values  are  smaller  than  the  corre 
sponding  coefficients  of  the  development  of* 

M 

z 

i 

r 
tn  a  series. 

This  may  also  be  written 

(5)  /«/—  (arguments). 


The  results  of  the  last  paragraphs  permit  a  simple  generaliza 
tion  to  the  case  for  which  another  point  of  the  plane  is  used  in  place 
of  the  origin.  If,  therefore,  f(z)  is  a  function  which  is  regular 
in  the  neighborhood  of  s=a,  it  is  transformed  by  the  substitution 

(6)  z-a  =  t> 

into  a  function  <£(£)  of  £,  which  is  regular  in  the  neighborhood 
of  the  origin,  and  hence  by  IV  can  be  developed  in  the  MAC 
LAURIN'S  series  : 


If  we  again  introduce  z  and/  in  place  of  £  and  <£,  we  obtain 

VI.    The  TAYLOR  series  : 
(7)         /(*)=/(*) 


*  The  number  M  is  definitely  defined  by  this  equation  just  as  it  is  when  re 
stricted  to  real  numbers  (A.  A.  I,  §  79)  ;  we  notice  also  that  the  M  thus  defined 
need  not  be  the  smallest  of  the  numbers  M  for  which  a  system  of  inequalities  of 
the  form  (4)  exists. 


210  IV.    SINGLE-  VALUED   ANALYTIC   FUNCTIONS 

It  converges  inside  of  a  circle  drawn  about  the  point  z  =  a  as  a 
center,  in  which  the  function  /(z)  is  regular. 

In  place  of  the  formulas  (3)  we  have  in  this  case  : 


(8) 


These  integrals  are  to  be  taken  along  a  curve  which  makes  a 
simple  circuit  about  the  point  z  =  a  and  which  belongs  entirely 
to  the  domain  in  which  the  function  is  regular.  The  first  of 
these  formulas  is  identical  with  (3),  §  36  ;  the  remaining  ones 
can  be  obtained  by  replacing  the  function  f(z)  in  that  formula 
by  its  derivatives  and  then  repeating  partial  integration. 

By  means  of  these  results  Theorem  I  may  be  generalized  fur 
ther  as  follows : 

VII.  When  two  functions,  of  z  regular  inside  of  a  domain  B 
coincide  along  an  arc  of  a  curve  however  small  belonging  to  this 
domain,  they  coincide  everywhere  in  the  domain. 

For,  if  a  be  a  point  on  an  arc  of  this  curve,  we  can,  for  the 
circle  of  convergence  K,  prove  the  coincidence  of  the  develop 
ments  of  both  functions  arranged  according  to  powers  of  z  —  a. 
If  there  are  points  of  B  outside  of  K,  we  can  then  find  a 
point  b  in  K  which  is  farther  distant  from  all  points  of  the  bound- 


§  39-  TAYLOR'S,  MACLAURIN'S  COMPLEX  POWER  SERIES      2 1 1 

ary  of  B  than  from  the  nearest  point  of  K.  Therefore,  the 
development  of  the  two  functions  in  powers  of  z  —  b  converges 
also  in  points  outside  of  K\  and  we  obtain  accordingly  their 
coincidence  for  these  points.  In  this  way  the  coincidence  of 
the  two  developments  can  be  proven  for  all  inner  points ;  for 
the  boundary  points,  it  follows  from  the  continuity.  Moreover, 
corresponding  conclusions  can  also  be  drawn  when  the  coinci 
dence  of  the  two  functions  is  known  only  for  the  points  of  a  set 
which  has  a  limit  point  on  t/ie  inside  of  the  domain  in  which  the 
functions  are  regular. 

We  return  at  this  point  to  Theorem  XII  of  the  previous  para 
graph,  which  is  at  once  applicable  to  series  in  powers  of  z  —  a. 
But  in  consideration  of  VI  it  can  be  expressed  as  follows : 

VIII.  If  a  point  ZQ  be  given  inside  the  domain  in  which  a  regular 
function  of  z,  f(z),  is  defined,  then  every  zero  of  this  function  differ 
ent  from  z0  is  distant  from  ZQ  by  more  than  an  assignable  quantity. 

Therefore,  the  zeros  of  a  regular  function  can  have  a  limit 
point  nowhere  within  the  domain  in  which  it  is  defined  (but  at 
most  on  its  boundary).  Accordingly,  on  account  of  XVI,  §  25, 
it  follows : 

IX.  In  every  domain  which  lies  entirely  within  the  domain  in 
which  a  regular  function  is  defined,  there  are  only  a  finite  number 
of  zeros  of  this  function. 

EXAMPLES 

1.  The  series  i  +  az  +  (rz*  +  -  •  •,  (a  >  o),  has  a  circle  of 
convergence  whose  radius  is  equal  to  -  •  How  does  the  series 
behave  inside  of,  upon,  and  outside  of  this  circle  of  convergence  ? 

NOTE.  —  Let  z  =  r\  be  any  point  on  the  positive  real  axis.  If  the  power 
series  converges  when  z  =  r\,  it  converges  absolutely  for  all  points  inside  the 
circle  |  z  \  =  r\  and,  in  particular,  for  all  real  values  of  z  less  than  r\. 


212  IV.    SINGLE-  VALUED   ANALYTIC   FUNCTIONS 

2.  What  is  the  radius  of  convergence  for  the  series 

£  +  ?!  +  *!  +  ... 

i2     22     32 
and  how  does  it  behave  at  all  points  on  its  circle  of  convergence  ? 

22        2s 

3.  The  series  z  ---  1  ---  •  •  •  has  its  radius  of  convergence 

2      3 

equal  to  unity.  It  diverges  for  z  =  —it  but  is  convergent 
(though  not  absolutely)  for  all  other  points  on  the  circle  of 
convergence,  since  its  real  and  imaginary  parts  are  cos  6  — 

COS29  +  ...  and  sin  0  -  ^^  +  •  •  -.     Give  the  proof. 

2  2 

4.  If  |  z  |  is  less  than  the  radius  of  convergence  of  either  of 
the  series  ^]anzn,  ^V/>nsn,  then  the  product  of  the  two  series  is 

"  when  cn  =  a$n  +  a^n-i  4-  #2^1  -2  +  *  *  '  ^/A-     Prove. 


5.  What  is  the  condition  for  the  convergence  of  the  product 
of  two  series  ?     (That  they  be  absolutely  convergent,  since  this 
permits  of  a  rearrangement  of  the  terms  in  any  order.     How 
ever,  two  series  not   absolutely  convergent  may  be   multiplied 
provided  only  that   the  product  is   convergent.       This    result  is 
known  as  ABEL'S  theorem  on  the    multiplication  of  series.) 

6.  If  the  radius  of  convergence  of  *^\anzn  is  r  and  f(z)  is 
the  sum  of  the  series  when  |  z    <  r  and    z  \  is  less  than  either  r 


i  —  z 


or  unity,  then     \      =      JM«"  where  Jn  =  tf0  +  #1  +  #2  -H  •  •  •  +  an. 


7.    Show  by  squaring  the  series  for  -      -  that 


2 

8.    Prove  in  the  same  way  that  -  —  ^—  =  i  +3  z  +  6  £  + 

(i  -  z)3 

the  general  term  being  \(n  -+-  1)(«  +  2)  •  zn. 


§  39-  TAYLOR'S,  MACLAURIN'S  COMPLEX  POWER  SERIES     213 
9.    lt/(z)  »!+,+  -!?-+...  show  that/(s)/00  =/(s  +  }'). 


[The  series  for/(z)  is  absolutely  convergent  for  all  values  of 


z  :  and  if  un  =  —  and  vn  =  ^,  it  follows  that  wn  =  i^    .] 
«  1  «  !  «  1 

10.  Expand  the  function  log  (i  -{-  e*)  to   five  terms  by  TAY 
LOR'S  theorem,  and  determine  the  radius  of  convergence  of  the 
series. 

11.  When  and  where  did  CAUCHY  first  publish  his  theorem 
about  the  extension  of  TAYLOR'S  theorem  to  functions  of  a  com 
plex  variable  ? 

12.  Given    j  -  *—  -  -  :  determine  the  domain  such  that 

J  (z  —  a)(z  —  I)) 

the  value  of  this  integral  taken  along  its  boundary  shall  be  equal 
to  zero. 


HINT.-  I  -, "     "~     .    =  f  ^;  =  /r*).2«=T^-.2«foracir- 


cle  center  at  b,  and  a  similar  expression  for  a  circle  center  at  a.     Their  sum 

=  a   ~ —  •  2  IT  i  and  this  =  O  for  certain  solutions,  a  =  b  excluded,  and  there- 
a—  b 

fore  the  configuration  may  be  obtained  accordingly. 

13.  Find  the  value  of   j       z     taken  along  a  circle  whose  cen- 

*/  z  —  i 

ter  is  z=  i  and  radius  <  2.  Why  not  take  the  radius  of  the 
circle  >  2  ? 

r*^z  .   jr 

14.  Evaluate    I  ^-— -  -  for  a  circle  having  its  center  at  i   and 

J  z  —  i 

radius  <  i  :  also  for  a  circle  having  its  center  at  i  and  radius 
>  i  and  <  2  :  also  its  value  for  a  circle  having  its  center  at  i 
and  radius  >2. 


214  IV.    SINGLE- VALUED   ANALYTIC   FUNCTIONS 

x%  7 

15.    Evaluate    I  ^os  7rZ  z  taken  along  a  circle  whose  center  is 
J  (z-  i)5 


-  7T5  -   i 


12 


(*-!)' 

the  origin  and  radius  >  i.  Ans. 

HINT.  — Apply  the  formula  /"(a)  =—   (    f\*)dz      along  any  contour 

2  7TZ  J    (z  —  «)n+1 

including  the  point  a, 

16.  Give  examples  of  power  series  which  converge  on  some, 
none,  and  all  points  of  their  circles  of  convergence. 

17.  Determine  the  circle  of  convergence  for  the  series 


-i)  .  ft  (ft 


I  •  2  •••  n  y  (y  -f    i)  •  •  •   (y  -f-  n  —  i) 

where  a,  /?,  y  are  not  negative  integers  and  y  =£  o.  This  series 
is  known  as  GAUSS'S  series  and  belongs  to  the  class  of  hyper- 
geometric  series.  See  GAUSS,  Ges.  Werke,  Vol.  Ill,  p.  125, 
RIEMANN,  Werke,  1876,  p.  79,  PIERPONT,  Functions  of  a  complex 
Variable,  p.  54. 

HINT.  —  By  the  ratio  test,  the  series  converges  absolutely  where  |  z  \  <  i  and 
diverges  for  |  z  \  >  i.  If  |  z  \  =  i,  it  converges  for  a  +  j3  —  y  <  o,  diverges  for 
a  +  /3  —  7  >  o. 

The  series  is  one  of  very  great  generality  and  includes  as 
particular  examples  many  well-known  series,  as 

18.  If  the  absolute  values  of  the  coefficients  of  the  integral 
power  series 

«=0 

are  finite,  the  circle  of  convergence  has  at  least  the  radius  i  ; 
when  is  it  exactly  i  ? 


§  39.   TAYLOR'S,  MACLAURIN'S  COMPLEX  POWER  SERIES     21  5 

19.  If  in  the  convergent  power  series 

f(z)  =  a0  +  a^z  +  ••• 

the  constant  term  aQ  is  not  zero,  a  number  k  can  be  found  such 
that/(s)  vanishes  for  no  value  of  z  whose  absolute  value  is  less 
than  k. 

20.  Develop  the  following  functions  of  z  in  an  integral  power 

series  in  z  : 

,  ,  i  —  z  cos  a  /2x   __  sjnj*  __ 

i-2scos«  +  s2'  i  -  2  -.  cos  «  +  s2' 

The  circle  of  convergence  of  these  power  series  has  the  radius 
unity  since  the  denominator  of  both  functions  vanishes  in  the 
points  z  =  cos  a±i  sin  a. 

21.  The  FRESNEL  integrals  are 


Obtain  power  series  in  z  for  C(z)  and  S(z).     Calculate 
C(i)  =  .72i7,  C(3)  =  .56io,  5(.i) 

7T 

22.    Show  that    |      sin*  (sXs  ==  .9309+. 
«/o 


23.    ^= 


\2'4  \2-4- 


Obtain  this  result.     The  time  of  swing  of  a  simple  pendulum 

of  length  /  through  an  angle  a  is  4\/-  •  K  where  /£  =  s 
Compute  this  time  when  «  =  60°, 


2l6  IV.    SINGLE- VALUED   ANALYTIC   FUNCTIONS 

§  40.    The  Exponential  Function  and  the  Trigonometric 
Functions,  Sine  and  Cosine 

In  the  first  chapter  we  inquired  about  combinations  of  com- 
plex  numbers  which  follow  the  same  fundamental  laws  as 
the  combinations  of  real  numbers  considered  in  elementary 
arithmetic  and  to  which  we  accordingly  applied  the  same 
names,  designated  them  by  the  same  symbols,  and  regarded 
them  as  generalizations  of  elementary  algebraic  operations. 
In  the  same  way  we  inquire  now  about  a  generalization 
of  the  transcendental  functions  of  real  argument  treated 
in  elementary  analysis  ;  for  the  simplest  of  these  functions, 
the  methods  already  deduced  are  sufficient  to  answer  this 
question. 

What  is,  for  example,  e*  (or  sin  z)  for  complex  values  of  z? 
In  itself  it  has  no  logical  meaning  whatever.  To  give  it  such  a 
meaning,  we  inquire  whether  there  exists  a  function  of  a  com 
plex  argument  z  which  has  the  same  properties  as  the  function 
of  a  real  variable  x,  designated  by  e*  in  the  elementary  theory, 
and  which  reduces  to  this  function  when  a  real  value  x  is  given 
to  z.  This  cannot  at  once  be  answered  in  the  affirmative  ;  for, 
properties  which  are  consistent  with  each  other  for  real  values 
of  z  may  be  contradictory  for  complex  values  (cf.  §  30).  We 
see  that  a  certain  freedom  is  unavoidable  here :  we  are  com 
pelled  to  retain  some  of  the  properties  of  a  function  of  real 
argument  in  order  to  use  them  as  the  basis  for  the  definition  of 
its  generalization  for  complex  values  ;  the  object  of  the  investi 
gation  then  is  to  find  out  which  of  the  remaining  properties  of  the 
given  function  of  real  argument  are  valid  for  such  generalization. 

Whatever  specially  concerns  the  functions  **,  sin  z,  cos  2,  they 
are  represented  in  elementary  analysis  (A.  A.  n,  §  71  ;  6,  §  75) 
by  the  power  series  : 


§4°.   EXPONENTIAL  AND  TRIGONOMETRIC  FUNCTIONS      2 1/ 


(3) 

It  is  easily  shown  that  these  series  converge  for  all  real  finite 
values  of  z.  According  to  I,  §  38,  they  then  converge  for  all 
finite  complex  values  of  z  and  represent  (§  38)  transcendental 
integral  functions  of  z.  Accordingly  : 

I.  There  are  three  transcendental  integral  functions  of  a  complex 
argument  z  represented  by  the  series  (-*")-( j),  whose  values  for  real 
values  of  the  argument  coincide  with  the  values  of  the  functions  e*  or 
exponential  z,  sin  z,  cos  z,  defined  in  elementary  analysis  ;  we  retain 
here  the  names  and  the  symbols  of  these  functions  of  a  real  argument. 

We  find  directly  from  the  definition  by  means  of  the   series 
(i)-(3)that: 

II.  The  following  relations  due  to  EULER  hold  between  these 
functions  : 

(4)  e"  =  cos  z  +  i  sin  z, 

(5)  e~iz  =  cos  z  —  i  sin  z 
with  their  solutions : 

(6)  cos  z  =  ^(tf"  +  <?-<•)» 

(7)  sin  *=-p (''"'- '"'*)» 

or  what  is  the  same  thing : 

(8)  cos /s=a(<f +  <?-•), 

(9) 


2l8  IV.    SINGLE- VALUED   ANALYTIC   FUNCTIONS 

III.  We  shall  make  particular  use  of  equation  (^)  in  order  to 
represent  a  complex  number  in    terms  of  its  absolute  value  and 
amplitude  (II,  §  4)  in  the  following  shorter  form  : 

(10)  *=*f*. 

A  fundamental  property  of  the  exponential  function  of  real 
argument  is  expressed  in  the 

IV.  Addition  theorem  : 

(n)  &+•*=&.*; 

we  may  verify  its  existence  for  complex  arguments  by  multiply 
ing  together  the  series  on  the  right  (A.  A.  §  60)  with  the  aid  of 
elementary  properties  of  the  binomial  coefficients.  By  repeated 
application  of  this  theorem  and  putting  all  arguments  equal  to 
each  other,  it  follows  that  the  equation : 

(12)  *"•  =  (*•)" 

is  true  for  integers  n,  understanding  the  exponents  on  the  right- 
hand  side  to  be  positive  integers  as  in  §  18.  If  z2=  —  z1,  it 
follows  from  (u)  that: 

(13) 


If,  further,  we  express  the  sine  and  the  cosine  of  a  sum  in 
terms  of  the  exponential  functions  by  (6)  and  (7),  apply  the 
addition  theorem  (n)  to  them,  and  then  introduce  the  trigono 
metric  functions  again  by  means  of  (4)  and  (5),  we  obtain : 

V.  The  addition  theorems  for  the  trigonometric  functions,  sine 
and  cosine: 

(14)  sin  (zl  +  z2)  =  sin  zl  •  cos  z2  -f  cos  zv  •  sin  z^ 

(15)  cos  (%  -f-  js2)  =  cos  Zi  •  cos  z2  —  sin  %  •  sin  z.2. 

From  the  second  of  these  equations  it  follows  for  z2=  —z^  that: 

(16)  cos2s-f  sin20  =  i. 


§40.    EXPONENTIAL  AND  TRIGONOMETRIC  FUNCTIONS      2IQ 

By  differentiation  of  the  series  (i)-(3)  term  by  term,  which  is 
permissible  according  to  IV,  V,  §  38,  we  obtain  : 

VI.    The  differential  equations  : 

(-7)         -* 


-       - 

We  have  thus  established  a  generalization  of  the  fundamental 
properties  of  the  real  functions  **,  sin  z,  cos  z  to  the  functions  of 
a  complex  argument  which  are  similarly  designated. 

EXAMPLES 

1.  Solve  the  equation  cos  z  =  a,  where  a  is  real. 

Put  z  —  x  +  />»  and  equate  real  and  imaginary  parts.  Thus 
cos  x  •  cosh  y  =  a  ;  sin  #  •  sinh  y  =  o  (where  cos  (/v)  =  cosh  jy 
and  sin  (ty)  =  *  sinh_y  by  definition,  §  40,  II).  Therefore  either 
y  =  o  or  x  is  a  multiple  of  TT.  If,  first,  y  =  o  then  cos  a;  =  0, 
which  is  impossible  unless  —  i  ^  a  <  i.  This  leads  to  the 
solution:  »  =  *  Ar  ±  COS  -><• 

wrhere  cos"1  0  lies  between  zero  and  ?r. 

If,  second,  x  =  m-  then  cosh  _y  =  (—  i)mtf,  so  that  either  a  >.  i 
and  w  is  even,  or  a  ^  —  i  and  m  is  odd.  If  a  =  ±  i  then  y=  o 
and  this  is  the  first  case.  If  \  a  \  >  i,  then  cos  iy  =  \  a  and  we 
obtain  the  solutions 

z  =  2  kir  ±/Log  \a  +  V^2  —  i  \,    (<z>i), 

z  =  (2  k+  i)7r±/Log  {-  a  +  V«2-i|,    («?<-!). 

2.  The  solution  of  cos  2  =  —  5/3  is  z  =  (2  k  +  I)TT  ±  /Log  3. 

3.  Solve  sin  z  =  a  where  a  is  real. 

4.  Solve  the  equation  tan  z  =  0  where  #  is  real.     (The  roots 
are  all  real.) 


220  IV.    SINGLE-VALUED   ANALYTIC  FUNCTIONS 

5.  Solve  the  equation  cos  z  —a  +  ib  (b  3=  o).  Let  us  take  b>o 
since  the  results  for  b  <  o  may  be  obtained  by  merely  changing 
the  sign  of  b.  For  this  case,  therefore, 

(i)  cos  x  •  cosh  7  =  a ;  sin  x  •  sinhy  —  —  b, 

and 

(tf/coshjy)2  +  (£/sinh_y)2  =  i  (using  the  notation  of  Ex.  i). 

The  solution  of  this  last  equation  gives 

A,  ±  A, 


where      Al  =  ^(a  +  i)2  +  P,   A2  =  £V(0  -  i)2  +  b\ 

Suppose  now  a  >  o.     Then  Al  >AZ  >o  and  coshy=A1  ±AZ. 
Moreover  cos  x  =  a/cosh  y  =  AI  T  A^ 

and  since  cosh  y  >  cos  x  we  must  take 

cosh  y  =  Al  -f-  A2  and  cos  x  —  Al  —  A2. 

The  general  solutions  of  these  equations  are 

(2)  x  =  2  kir  ±  cos~lM,  y  =  ±  Log  \L  +  V^2  —  i  \ 
where  L  —  A±  +  Az,   M=Al  —  A2  and   cos~lM  lies  between 
zero  and  -. 

2 

The  values  of  x  and  7  thus  found  include  the  solutions  of  the 
equations 

(3)  cos  x  •  cosh  y  =  a,    sin  x  -  sinh  y  —  b 

as  well  as  those  of  equations  (i),  since  we  have  used  only  the 
second  of  equations  (3)  after  squaring  it.  To  distinguish  the 
two  sets  of  solutions  we  observe  that  the  sign  of  sin  x  is  the 
same  as  the  ambiguous  sign  in  the  first  of  equations  (2),  and 
the  sign  of  sinhjy  is  the  same  as  the  ambiguous  sign  in  the 
second.  Since  b  >  o  these  two  signs  must  be  different.  Hence 
the  general  solution  required  is 

z  =  2k  ±  [cos-1  M—  /Log  \L  +VZ2  —  ij]. 


§4o.    EXPONENTIAL  AND  TRIGONOMETRIC  FUNCTIONS      221 

6.  Study  the  cases  of  the  last  problem  where  a  <  o  and  a=o. 

7.  Show  as  in  Ex.  5  that,  if  a  and  b  are  positive,  the  general 
solution  of  sin  z  =  a  +  ib  is 


s  =  k*  +  (-  i^sin-1  J/+  /Log  \L  +  VZ2  -  1 }], 
where  sin"1  J/ lies  between  o  and  -. 

2 

8.    Show  that  the  general  solution  of  tan  z  =  a  +  ib,  b  =£  o,  is 

0         /  (V    +(j    +  W] 

2  =  /C-7T  4-  -  +  _  Log  \  -YV7—  ^  }• 
2      4  L<Z  +  (i  -  *)*  ] 

where  0  is  the  numerically  least  angle  such  that 


cos  0  :  sin  0  :  i  :  :  i  -  a2-  -  P  :  2  a  :  V(i  -  a2  -  £*)2  +  4  rt2. 

9.  Calculate  cos  /,  sin  (i  +  /'),  sin  (n-  —  i  —  i)  to  two  places 
of  decimals  by  means  of  the  power  series  for  cos  z  and  sin  z. 

10.  Prove  that    cos  z  \  ^  cosh    z  \  and  |  sin  z     5^  sinh  |  z  \  . 

11.  Show  that  |  cos  z  \  <  2  and  |  sin  z  \  <  6/5  J  z  \  if  |  z  \  <  i. 

12.  Since  sin  22  =  2  sin  z  •  cos  5  we  have 


----  -_          -----      -— 

3!         5!  I       3!     S!         A      a!T 

Prove  by  multiplying  the  two  series  on  the  right-hand  side  and 
equating  coefficients  that 


for  which  the  notation  fw>)  means  ;/'  ;/  ~  Im  "'  n  ~  r  +  I.    Verify 
\r)  i.  2.  •••  r 

the  result  by  the  binomial  theorem. 


222  IV.    SINGLE- VALUED   ANALYTIC   FUNCTIONS 

§  41.     Periodicity  of  the  Trigonometric  and  the  Exponential 
Functions 

The  sine  and  the  cosine  of  a  real  argument  are  periodic* 
functions  with  the  period  2  TT  ;  that  is,  the  following  equations  : 

(1)  sin  (z  +  2  TT)  =  sin  2,    cos  (z  +  2  TT)  =  cos  z, 

are  identically  true  for  all  real  values  of  z  (A.  A.  §  76).  We 
can  at  once  conclude  from  this  and  from  Theorem  I,  §  39,  that 
these  equations  must  also  hold  for  all  complex  values  of  z. 

(Periodic  functions  are  a  special  kind  of  automorphic  func 
tions  (IV,  §  17)). 

It  then  follows  from  the  relations  due  to  EULER  that  • 

(2)  ?+*«*  =  ?, 
that  is, 

I.  The   exponential  function   is   a  periodic  function   with   the 
period  2  tri. 

It  is  further  shown  in  the  theory  of  trigonometric  functions 
of  real  arguments  (A.  A.  VI,  §  76)  that  2  ?r  is  a  primitive  period 
of  the  cosine  and  of  the  sine  ;  that  is,  that  no  aliquot  part  of  2  IT 
is  a  period  of  these  functions.  It  thus  follows  indirectly  from 
equation  8,  §  40,  that : 

II.  2  iri  is  a  primitive  period  of  the  exponential  function. 

We  shall  now  investigate  whether  the  exponential  function 
has  other  primitive  periods  besides  2  nt  and  —  2  tri.  For  this 
purpose  we  deduce  the  following  theorems  from  its  definition 
and  from  equations  (13)  and  (17)  of  §  40  (cf.  also  A.  A.  §  52) : 

III.  The  exponential  function  increases  continuously  from  o  to 
-f-  oo,  while  its  argument  continuously  increasing  takes  on  all  real 
values  from  —  oo  to  -f-  oo. 

*  Cf.  Ex.  4,  following  §  18,  and  Ex.  31  at  the  end  of  chap.  IV.  —  S.  E.  R. 


§41.   PERIODICITY   OF  TRIG.   AND   EXP.   FUNCTIONS       223 

IV.    //  takes  on  therefore  each  real  positive  value  for  one  and  for 
only  one  real  value  of  its  argument. 

Assume  now  that  a  is  a  period  of  the  exponential  function 
and  that  z±  and  22  are  two  values  differing  by  a  so  that 

(3)  z,-2l  =  a] 
then 

(4)  *  =  *. 

By  means  of  (4)  and  (n)  of  §  40  we  can  separate  the  real  and 
imaginary  parts  of  the  exponential  function  ;  thus 

(5)  <?z+iy  =  e*  cos  y  +  /  ex  sin  y. 
If  we  then  put      zl  =  A\  -f  iy\,   z*  =  x2  -h  iy& 

it  follows  from  (4)  that 

(6)  e*i  cos  }\  =  e1*  cos  j,.    **'  sin  )\  =  ^  sin  _y2, 

and  from  both  equations  on  account  of  (16),  §  40  : 


But  from  this  it  follows  according  to  IV,  that 

xl  =  x2, 

and  from  the  properties  of  trigonometric  functions  of  real  argu 
ments  (A.  A.  p.  167,  line  7)  : 

J>'2  =  J'i  +  2  kir. 

We  thus  obtain  the  theorem  : 

V.  Every  period  of  the  exponential  function  is  an  integral  mul 
tiple  0/2  Tri  and  every  period  of  the  sine  or  the  cosine  is  an  integral 
multiple  of  2  TT. 


224  IV-    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

We  add  also  the  following  definition : 

VI.  A  periodic  function,  all  of  whose  periods  are  integral  mul 
tiples  of  a  primitive  period,  is  called  a  singly  periodic  function. 

We  can  therefore  state  the  theorem : 

VII.  The  exponential  function,  the  sine,  and  the  cosine  are  singly 
periodic  functions. 

§  42.    Conformal  Representations  Determined  by  Singly  Periodic 

Functions 

The  conformal  representations  determined  by  the  functions 
e*,  sin  z,  cos  z  can  now  be  investigated  in  detail  by  means  of  the 
results  of  the  preceding  paragraph.  For  the  first  of  these 
functions  it  therefore  follows  from  those  results  that: 

I.  The  function  w  =  e*  takes  on  every  finite  value  w,  different 
from  zero,  at  an  infinite  number  of  points  of  the  z-plane,  all  of 
which  follow  from  any  one  of  them  by  addition  and  subtraction  of 
arbitrary  integral  multiples  of  2  -n-i. 

Let  us  draw  then  in  the  £-plane  two  parallels  to  the  ^-axis  at 
a  distance  2  ?r  from  each  other  and  regard  one  of  these  lines  as 
belonging  to  the  strip  bounded  by  them,  the  other  not ;  in  this 
way  each  point  of  any  such  set  of  points  occurs  just  once  in  the 
strip.  Hence  every  such  strip  is  mapped  exactly  upon  the  whole 
w-plane  by  the  function  w  =  e*\  by  the  general  theorem  of  §  34 
the  representation  is  conformal.  Thus,  in  the  terminology 
introduced  in  VI,  §  17,  we  say: 

II.  Every  strip  bounded  by  two  lines  parallel  to  the  x-axis  and 
drawn  at  a  distance  2  TT  from  each  other  can  be  looked  upon  as  a 
fundamental  region  of  the  function  e*.      We  shall  call  such  a  strip 
a  period  strip  of  the  function. 

Besides,  it  follows  from  its  definition  in  terms  of  a  permanently 
converging  power  series  with  real  coefficients,  that  the  function  f 


§42.   MAPPING   WITH   SINGLY   PERIODIC  FUNCTIONS       225 

has  the  property  that  it  takes  on  real  values  for  real  values  of  s,  and 
conjugate  imaginary  values  for  conjugate  imaginary  values  of  z\ 
it  is  therefore  a  symmetric  automorphic  function  in  the  sense  of 
the  definition  given  in  X,  §  18.  Accordingly,  we  can  regard  two 
pieces  of  the  period  strip  symmetrical  to  each  other  with  refer 
ence  to  the  .r-axis  as  its  fundamental  region.  This  is  seen  to 

z-p/ane 

^  w-pfane 


^WVVVVVVVV 

AAAAAAAAAAA^ 

FIG.  19 

be  true  in  this  case  if  we  bound  the  strip  by  the  two  straight 
lines  y  —  ir  and  y  =  —  TT.  If  we  map  this  strip  upon  the  w-plane 
by  w  =  e*,  it  will  have  a  cut  along  the  negative  real  axis  and  the 
two  "  banks  "  of  this  cut  will  correspond  to  the  two  borders  of  the 
strip.  This  is  shown  in  Fig.  19  and  in  the  former  figures  10,  n. 
We  determine  also  the  curves  of  the  w-plane  which  correspond 
to  the  parallels  to  the  axes  of  the  s-plane  : 

III.  From  the  conformal  representation  determined  by  w  =  e* 
we  obtain  the  following  results :  To  the  parallels  to  the  y-axis 
(x  =  consfy  correspond  the  circles  of  the  w-plane : 

(1)  //2+P*=^ 

whose  radii  increase  in  geometrical  progression  while  the  abscissas 
of  the  straight  lines  increase  in  arithmetical  progression ;  to  the 
Parallels  to  the  x-axis  ( y=-const?)  correspond  the  rays  of  the  w-plane: 

(2)  u  sin  y  —  v  cos  y  =  o 

which  form  equal  angles  with  each  other  providing  the  straight  lines 
parallel  to  the  x-axis  follow  each  other  at  equal  distances. 


226 


IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 


For  the  functions  sine  and  cosine,  we  also  have  period  strips 
of  breadth  2  TT  ;  however,  in  this  case,  the  strips  are  bounded 
by  parallels  to  the  jy-axis  as  for  eiz.  But  while  eiz  takes  on  every 
value  in  such  a  strip  once,  sine  and  cosine  take  on  every  value  in 
it  twice.  This  follows  from  the  fact  that  these  functions  have 
transformations  into  themselves  other  than  their  periodicity  as 
the  following  equations  show  : 

(3)  sin  (TT  —  z)  =  sin  z 

(4)  cos  ( —  z)  =  cos  z. 

But   they  cannot   take  on  this  same  value  oftener,  since,  for 
example : 

(5) 


is  a  rational  function  of  the  second  degree  in  eiz  and  hence 
(VII,  §  20)  cannot  take  on  one  and  the  same  value  for  more 
than  two  values  of  eiz. 

z-p/ane 


// 


FIG.  20 

The  period  strip  is  in  this  case,  therefore,  not  a  fundamental 
region;  but,  since  sine  and  cosine  are  symmetric  automorphic 
functions,  we  obtain  such  a  fundamental  region  for  the  cosine 
according  to  XI,  §  18,  by  bounding  it  by  x  =  o  and  x  =  TT  ;  cf. 
Pig.  20.  The  w-plane  in  this  representation  has  a  cut  from  —  i 


§43-    POLES  OR  NON-ESSENTIAL  SINGULAR   POINTS       22/ 

to  —  oo  and  from  +  i  to  +  oo.     The  strip  bounded  by  x  = 

and  x  =  4-  ^  is  a  fundamental  region  for  the  sine. 

If  we  put 

(6)  u  +  iv  =  cos  (.v  +  /», 

we  obtain 

gv    i    g-y  gv g— 9 

(7)  //  =  —     —  •  cos  x,    v  = —  •  sin  x, 

and  hence 


(  "  V    (  *  V 

\cos  x)       \^sin  .xy 


=  i  ;    that  is, 


IV.  In  the  map  determined  by  w  =  cos  z,  parallels  to  the  axes 
of  the  z-plane  correspond  in  the  w-plane  to  confocal  ellipses  and 
hyperbolas  whose  foci  are  at  the  points  ±  i. 

§  43.    Poles  or  Non-essential  Singular  Points 

Up  to  this  time  we  have  limited  the  investigation  of  functions 
of  a  complex  argument  to  such  domains  in  which  the  function 
to  be  investigated  was  regular ;  we  proceed  now  to  the  investi 
gation  of  the  case  for  which  there  are,  in  the  domain  under 
consideration,  particular  points  to  be  excepted,  at  which  there  is 
either  no  value  of  the  function  given  originally,  or  the  given 
value  of  the  function  in  its  relation  to  the  adjacent  values  no 
longer  satisfies  all  the  hypotheses.  Let  us  fix  in  mind  then  one 
such  exceptional  point ;  without  loss  of  generality  we  may  sup 
pose  that  it  is  the  point  z  =  o. 

The  simplest  case  would  be  where  the  function  can  be  made 
to  satisfy  all  the  conditions  in  the  neighborhood  of  the  origin, 
itself  included,  by  changing  the  value  which  the  function  takes  on 


228  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

at  the  origin  (or  by  selecting  such  a  value  when  according  to  the 
original  definition  the  function  has  no  definite  value  for  the 
value  z  =  o  of  the  argument).  We  then  say  :  the  given  value  of 
the  function  exhibits  a  removable  discontinuity  at  the  origin. 

An  example  *  of  this  is  furnished  by  the  rational  function  of 
§  20,  which  was  given  in  such  form  that  numerator  and  denomi 
nator  had  an  additional  factor  dependent  on  z.  We  may  ex 
clude  such  removable  discontinuities,  and  will  do  so  in  what  follows, 
by  supposing  the  original  definition,  if  it  included  such  singularities, 
to  be  already  modified  or  supplemented  accordingly. 

Furthermore,  we  have  already  become  acquainted  with  a 
definite  kind  of  singular  points  of  rational  integral  functions  to 
which  we  gave  the  name  pole.  Hence  the  following  general 
definition : 

I.  IVhen  a  function  f(z)  of  a  complex  variable  is  regular  in  the 
neighborhood  of  a  point  z  =  o,  the  point  itself  excluded ;  and  when 
further  an  integer  n  can  be  found  such  that  the  product : 

(0  a"  •/(*)  =/i(«) 

can  be  made  a  function  regular  at  z  =  O  BY  ASSIGNING  TO  IT 

AT  Z  =  0  A  DEFINITE  FINITE    VALUE  DIFFERENT  FROM  ZERO, 

then  we  say  that  z  =  o  is  a  pole  t  of  f(z)  of  order  n. 

The  reciprocal  of  such  a  function  is  in  general  not  defined 
at  the  point  z  =  o  ;  but  it  follows  that : 

II.  If  we  assign  the  value  zero  to  the  reciprocal  function  — —  at 

f\z) 

the  point  z  =  o,  there  is  defined  in  this  way  a  function  regular  in  a 
certain  neighborhood  about  the  point  z  =  o,  this  point  itself  included. 

*  For  other  illustrations  of  removable  discontinuities  see  examples  8  and  9,  at 
the  end  of  \  47.  —  S.  E.  R. 

t  According  to  WEIERSTRASS  "  ausserwesentlich  singularer  Punkt,"  that  is, 
"  non-essential  singular  point." 


§  44.    BEHAVIOR   OF  A   FUNCTION   AT   INFINITY         2  29 

According  to  XII,  §  38,  we  can  assign  a  neighborhood  about 
the  point  z  =  d  in  which  f^(z)  is  everywhere  different  from  zero 
and  therefore  —  ^—  is  regular.  Accordingly, 


is  regular  there. 

Since  f^z)  can  be  developed,  according  to  III,  §  37,  in  a 
series  of  the  form  : 

/iOO  =  "o  +  *is  +  •"  +«VM-  — 
it  follows  that  : 

III.  A  function  f(z),  which  has  a  pole  of  order  n  at  z  =  O,  has 
a  development  in  this  neighborhood  of  the  form  : 

(2)   f(z}  =  a.z'n  +  a,z~n+l  +  -+an-^  +  an  +  an+iZ  +  •••• 

Further,  from  this  and  XII,  §  38,  we  have  : 

IV.  About  every  pole  of  a  function  f(z),  we  can  draw  a  circle 
of  so  small  a  radius  that  neither  another  pole  nor  a  zero  of  the 
function  lies  in  it. 

§  44.    Behavior  of  a  Function  of  a  Complex  Argument  at  Infinity. 
The  Fundamental  Theorem  of  Algebra 

In  order  to  investigate  the  behavior  of  a  function  f(z)  of  a 
complex  argument  z  at  infinity,  let  us  transfer  the  neighborhood 
of  the  point  oo  of  the  s-sphere  to  the  neighborhood  of  the  zero- 

point  of  the  s'-sphere  by  means  of  the  substitution  z'  =  -,  z  =  - 

z  z 

as  in  the  special  case  of  rational  functions  (§  21).  and  consider 
/(z)  =  /(V)  =  <t>(s')  as  a  function  of  z1.     Thus  : 

I.  The  expression,  "  A  function  f(z)  has  such  and  such  a  prop 
erty  at  infinity  "  means  that  <j>(z')  =/(^\  considered  as  a  function 
of  z\  has  this  property  in  the  neighborhood  of  the  point  z'  =  O. 


230  IV.    SINGLE- VALUED   ANALYTIC   FUNCTIONS 

For  z'  =  o  itself,  this  does  not  yet  define  the  symbol  <j>(z'), 
but  when  it  is  possible  to  make  <f>(z')  regular  in  the  neighbor 
hood  of  the  origin  by  a  suitable  selection  of  a  value  for  <£(o)  (cf. 
the  previous  paragraphs),  then  we  say,  /(z)  is  regular  at  infinity. 
From  this  definition  and  from  III,  §  37,  it  follows  directly  that: 

II.  A  function  regular  at  infinity  can  be  developed  in  a  series  : 

(1)  f(z)  =  a0  +  a,z~l  +  a2z~*  +  •••  +  anz~n  +  - 

of  powers  of  z  with  negative,  integral,  decreasing  exponents,  which 
converges  absolutely  outside  of  a  certain  circle  with  z  =  o  as  a 
center.  Conversely,  such  a  series  always  represents  a  function 
regular  at  infinity. 

Likewise,  we  obtain  the  following  theorem  from  III,  §  43  : 

III.  If  a  function  has  a  pole  at  infinity,  it  can  be  developed  in  a 
series  of  the  form  : 

(2)  /O)  =  a_nzn  +  a_n+lzn~l  +  -.  +  a_^  +  a_,z  + 

a0  -f  a±z~l  +  a2z~*  -f  •••  +  anz~n  4-  •••  • 

We  now  take  up  a  theorem  due  to  LIOUVILLE  which  is  fun 
damental  for  all  further  detailed  investigations  in  function 
theory : 

IV.  A  function  of  a  complex  argument  regular  over  the  entire 
sphere  is  necessarily  a  constant. 

If  f(z)  is  regular  at  infinity,  there  is  then  a  quantity  J/i  hav 
ing  the  property  that  \f(z)  \  <  J/1}  whenever  |  z  is  greater  than 
a  definite  number  R.  If  f(z)  is  regular  everywhere  except  at 
infinity,  then  Theorem  V  of  §  39  is  applicable.  But  the  number 
M  appearing  in  that  theorem  is  <  J/i,  whenever  r>  -R ;  the 
coefficients  an  of  the  development  of  /(z)  in  a  MACLAURIN'S 
series  must  then  be  smaller  than  M^ .  r  ~n  for  any  value  of  r  how- 


§44-    THE   FUNDAMENTAL  THEOREM   OF  ALGEBRA       231 

ever  large.  But  that  is  not  possible  for  any  positive  value  of  // 
while  an  is  not  equal  to  o.  Therefore  f(z)  must  reduce  to  a0. 

Theorem  IV  is  thus  important  in  connection  with  functions 
of  which  only  the  properties  but  no  analytic  expressions  are 
known,  since  we  are  frequently  able  to  find  such  expressions 
with  the  help  of  this  theorem.  The  two  following  theorems  are 
the  first  examples  of  this  : 

V.  A  function,  which  is  regular  everywhere  except  at  infinity 
and  has  an  n-fold  pole  at  infinity,  is  a  rational  integral  function  of 
the  nth  degree. 

If  it  has  an  w-fold  pole  at  infinity,  a  development  of  the  form 

(2)  is  possible  there.     Thus  if  we  put 

(3)  VO)  =  a-n?  +  a_H+lsr~l  +  —  +  a_&  +  a, 

and  form  the  difference  f(z)  —  $(z)<  it  is  regular  everywhere 
except  at  infinity  ;  for,  f(z)  is  regular  according  to  hypothesis 
and  \l^z)  according  to  §  31.  But  it  is  regular  also  at  infinity 
according  to  II  and  is  therefore  a  constant  according  to  IV,  and 
in  fact  =  o,  since  it  is  equal  to  zero  for  z  =  oo.  Therefore  f(z) 
is  equal  to  the  rational  integral  function  \j/(z).  Q.E.D. 

VI.  A  function  '/(z)  which  is  regular  everywhere  over  the  whole 
sphere  with  the  exception  of  a  finite  number  of  poles  is  a  rational 
function. 

Let  av(y  =  i,  2,  ...  n)  be  the  poles  of/(z)  on  the  finite  part  of 
the  sphere,  kv  their  order  ;  let 


be  the  terms  with  negative  exponents  in  the  development  in  a 
series  valid  for  the  neighborhood  of  av  (III,  §  43).     If  we  form 


232  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

then  the  rational  function  : 


the  difference  f(z)  —  $(z)  is  regular  everywhere  except  at  infin 
ity,  even  at  the  points  ait  a2,  •••,  an;  at  infinity,  it  has  a  pole  or 
is  regular  according  as  f(z)  has  the  one  or  the  other.  It  is, 
therefore,  a  rational  integral  function  %(z)  according  to  V  or  a 
constant  (a  rational  integral  function  of  zero  degree)  according 
to  IV.  Therefore,  f(z)  is  equal  to  the  rational  function 


If  we  apply  this  theorem  to  the  reciprocal  of  a  rational  integral 
function  of  the  mth  degree  g(z\  x(z)  *s  m  anY  case  a  constant  ; 
and  if  we  next  reduce  \f/(z)  +  x(z)  *o  a  common  denominator,  a 

quotient  of  two  polynomials  -*J-r  is  obtained  whose  denominator 

/i.2(z) 

hz(z)  is  of  the  degree  k=kl-\-k%-\-  •••  -\-kn  and  whose  numerator 
HI(Z)  is  at  most  of  this  degree.     From  the  equation  : 

(6)  -i-^M*), 

*«      AiW 

or  h2(z)  =  hl(z^g(z), 

it  follows  then  that  £  must  be  ^  ;».     Thus  the  number  of  poles 

of  -*—  ,  that  is,  the  number  of  zeros  of  g(z)  (each  counted  as 

£<*) 
often  as  its  order  indicates),  is  at  least  equal  to  m  ;  and  since 

it  cannot  also  be  greater  than  m  according  to  II,  §  19,  we  have 
proved  the  fundamental  theorem  of  algebra  : 

VII.  Every  algebraic  equation  of  the  mth  degree  has  exactly  m 
roots  in  the  field  of  complex  numbers  of  the  form  a  +  bi,  where  mul 
tiple  roots  are  counted  according  to  their  order  of  multiplicity.* 

*Cf.  GORDAN'S  proof,  Invarianten,  Vol.  I,  p.  166  ;  also  OSGOOD,  Lehrbuch  der 
Funktionentheorie,Vo\.  I,  p.  185;  BOCHER,  Am.  Jour,  of  Math.,  Vol.  17  (1895), 
p.  260,  and  Bull.  Amer.  Math.  Soc.,  ad  Series,  Vol.  I  (1895),  P-  2O5  I  GOURSAT- 
HEDRICK,  Mathematical  Analysis,  Vol.  I,  pp.  3,  131,  291.  —  S.  E.  R. 


§  44-    BEHAVIOR   OF   A   FUNCTION   AT   INFINITY         233 

Let  us  inquire  now  how  a  transcendental  integral  function 
behaves  at  infinity.  We  can  at  present  answer  this  question  in 
part  from  the  behavior  of  the  singly  periodic  functions  investi 
gated  in  §§  40-42.  If  we  draw  about  the  origin  a  circle  with  a 
radius  however  large,  an  infinite  number  of  parallel  strips  will 
always  remain  entirely  outside  of  it;  therefore  a  periodic 
function  takes  on  every  value  which  it  takes  on  at  all,  an  infinite 
number  of  times  in  any  arbitrary  neighborhood  of  the  point  oo . 
For  example,  e*  takes  on  every  value  an  infinite  number  of 
times  in  any  arbitrary  neighborhood  of  the  point  oo  ,  excepting 
alone  the  two  values  o  and  oc  .  But  it  also  approaches  arbitra 
rily  near  to  these  two  values  in  any  neighborhood  of  the  point 
oo  ,  however  small. 

We  show  now  that  the  behavior  of  every  transcendental 
integral  function  at  infinity  is  similar  to  that  of  e*  ;  and  that  every 
such  function  takes  at  infinity  values  arbitrarily  large  in  addition 
to  other  values  ;  or  more  precisely  : 

VIII.  Given  a  transcendental  integral  function  f(z]  and  a  posi 
tive  number  M  (however  large),  tJiere  are  then  always  values  of  z 
outside  of  every  circle  (with  a  radius  arbitrarily  large),  for  which 

\A*)\>*f-  ' 

If  that  were  not  the  case  outside  of  a  circle  of  radius  ./?,  then, 
by  applying  Theorem  V,  §  39,  for  a  circle  of  radius  r  >  tf,  we 
could  prove  that  the  coefficients  in  the  MACLAURIN  development 
of  f(z)  are  respectively  smaller  than  Mr~*.  But  from  that  it 
would  follow  as  in  the  proof  of  IV,  that  they  must  all  be  zero. 

The  transcendental  integral  functions  thus  share  this  prop 
erty  with  the  rational  integral  functions  ;  but  they  differ  from 
them  as  follows  : 

IX.  Given  a  transcendental  integral  function  f(z)  and  a  positive 
number  c  (however  small),  then  there  are  always  points  outside  of 


234  IV-    SINGLE- VALUED   ANALYTIC   FUNCTIONS 

every  circle  (with  a  radius  arbitrarily  large),  at  which  f(z)  is 
SMALLER  than  e. 

This  is  self-evident  if  there  are  always  zeros  of  f(z)  outside 
of  every  circle.  But  if  all  the  zeros  of  f(z)  lie  inside  of  a  circle  of 
radius  7?,  there  can  be  only  a  finite  number  of  them  according 
to  IX,  §  39,  and  we  designate  them  then  by  aiy  a2,  •••,  an.  These 
points  are  poles  of  the  function  *  ;  as  in  the  proof  of  VI  (equa 
tion  4),  let  $v(z)  be  the  sum  of  the  terms  with  negative  exponents 
in  the  development  of  this  function  in  a  series  valid  for  the  neigh 
borhood  of  av.  It  then  follows  here  as  there,  that 

n 

(7)  —  -y«w*) 

f(z]         ; 
J  ^  '      v  =1 

is  everywhere  regular  except  at  infinity.  This  difference  is  then 
a  transcendental  integral  function  which  cannot  be  reduced  to  a 
constant  since  otherwise  f(z]  would  be  a  rational  function,  con 
trary  to  the  hypothesis.*  According  to  VIII,  therefore,  this 
difference  takes  on  values  arbitrarily  large  outside  of  every 
circle ;  since  every  $v(z),  and  therefore  their  sum,  becomes 

indefinitely  small  at  infinity,  it  follows  that  becomes  arbi- 

/  (*) 

trarily  large  there;  that  is,  f(z)  takes  on  values  arbitrarily 
small  at  infinity.  Q.E.D. 

If  Theorem  IX  be  applied  to  f(z]  —  c,  where  c  designates  an 
arbitrary  constant,  we  have  the  following  general  theorem : 

X.  A  transcendental  integral  function  approaches  arbitrarily 
near  to  every  value  in  the  neighborhood  of  the  point  oo  . 

We  must  not  understand  this  theorem  to  mean  that  such  a  func 
tion  actually  takes  on  every  value  in  the  neighborhood  of  z  =  oo  . 

*  That  a  rational  fractional  function  cannot  at  the  same  time  be  a  transcendental 
integral  function  follows  from  Theorem  VII. 


§44-    THE   FUNDAMENTAL  THEOREM  OF  ALGEBRA      235 

This  is  shown  by  the  exponential  function,  which  is  neither  o  not 
oo  at  any  assignable  point.* 

The  definition  of  a  transcendental  integral  function  fails  for 
f(s)  =  ao.  On  account  of  Theorem  X  there  is  no  object  in  try 
ing  to  complete  this  definition  to  conform  with  I,  §  21,  by  giving 
it  a  definite  value  even  at  z  =  oo  .  On  the  contrary,  it  is  possible 
at  times  to  obtain  a  definite  value  in  the  limit  when  the  variable 
z  is  allowed  to  approach  the  point  oo  along  an  assigned  path. 
Thus,  for  example,  <?*  converges  to  oo  when  z  approaches  oo 
along  the  axis  of  positive  real  numbers  and  converges  to  zero 
when  z  approaches  oo  along  the  axis  of  negative  real  numbers. 
On  the  other  hand,  its  real  part  as  well  as  its  imaginary  part 
fluctuates  continually  between  —  i  and  -f  i  when  z  approaches 
infinity  through  purely  imaginary  values,  positive  or  negative. 

EXAMPLES 

1.  Expand  — - —  for  the  domain  at  infinity;  that  is,  in  powers 

z  —  i 
of  -  valid  for  the  domain  outside  of  the  circle  whose  center  is 

2 

at  z  =  o  and  radius  i.     (Cf.  Ill,  §  43,  II,  §  44.) 

2.  Expand  (2+I>  (*+i)(*+ 2)    for   the   domain 

(,+  2)(*  +  3)'    (z  +  3)(z  +  4) 
at  infinity. 

3.  What  are  the  poles  of — ?    Expand  this  function  in 

a  circle  about  the  point  2  =  0.     Also  in  a  circle  about  z  =  i . 

4.  Expand  *(*+  ')  as  in  III,  §  44. 

z+  2 

*  PICARD  has  shown  (Par.  C.  R.  90,  1879)  that  there  is  never  more  than  one 
finite  value  which  will  not  really  be  assumed  by  a  transcendental  integral  func 
tion  in  the  neighborhood  of  z  =  oo  .  A  proof  of  this  theorem  by  elementary  means 
is  given  by  E.  BOREL,  Par.  C.  R.  122,  1896,  and  Lefons  sur  les  fonctions  entieres, 
Paris,  1900,  p.  103. 


236  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

5.    Give  the    domain  of    convergence   for   the    expansion  of 


sin  z 


Find  this  expansion. 


z  + 

6.    What  is  the  domain  of  convergence  for  the  expansion  of 
sm  z   in  powers  of  z  +  2  ?     Give  the  expansion. 


§  45.    Cauchy's  Theorem  on  Residues 

We  became  acquainted  in  IV,  §  35,  with  the  theorem  that  the 
value  of  the  integral 

"f(t)A 


is  always  equal  to  zero  if  it  is  taken  along  the  boundary  of  a 
domain  in  which  the  function  f(z]  is  regular  ;  we  now  inquire 
as  to  the  value  of  this  integral  when  a  finite  number  of  poles 
lie  in  the  domain. 

Let  us  consider  first  a  domain  in  which  there  is  only  one  pole, 
and  let  it  be  at  the  point  z  =  o.  According  to  VII,  §35,  we 
can  then  deform  the  path  of  integration  into  a  circle  about  the 
point  z  =  o  with  a  radius  arbitrarily  small  without  changing  the 
value  of  the  integral.  But,  according  to  III,  §  43,  in  the  neigh 
borhood  of  the  point  z  =  o, 

/CO  =  a-nz-*  +  a_n+}z-n+l  +  ...  +  «_,*-»  +  «_1r-1  4-/iO)> 

where/  (z)  designates  a  function  regular  in  the  neighborhood  of 
the  point  z  =  o  ;  we  thus  obtain  : 


all  of  these  integrals  being  taken  along  the  given  small  circle. 
The  last  one  of  these  integrals  is  zero  according  to  a  previous 
theorem  (IV,  §  35),  the  others  have  already  been  evaluated  in 


§  45-    CAUCHY'S  THEOREM    OX    RESIDUES 


237 


VIII,  §  35.     Introducing  here  the  values  so  found,  we  obtain  : 


Definition  : 

I.  The  coefficient  of  the  (  —  i)*  power  of  (z  —  a}  in  the  develop 
ment  of  a  function  in  the  neighborhood  of  t/ie  pole  z  =  a  is  called  the 
residue  of  the  function  at  this  pole. 

Accordingly,  equation  (i)  can  be  stated  as  follows: 

II.  The  integral    \  f  (z)dz,   taken  around  a  domain  in  which 

the  function  is  regular  with  the  exception  of  a  pole,  is  equal  to  2  iri 
times  tJie  residue  of  the  function  at  this  pole. 

But  if  we  have  a  domain  in  which  several  poles  lie,  we  divide 
it  into  a  number  of  subdomains  such  that  each  of  them  contains 
only  one  pole,  apply  The 
orem  II  to  each  of  these 
subdomains,  and  add  the 
results.  We  thus  inte 
grate  twice  along  each 
dividing  line  between 
subdomains  but  in  op 
posite  directions  (and 
always  so  that  the  sub- 
domain  under  considera 
tion  lies  to  the  left).  The 
integrals  taken  along  the  inner  boundaries  accordingly  cancel 
out  entirely  (cf.  VII,  §  29)  and  only  the  integral  along  the  out 
side  boundary  of  the  given  domain  remains.  We  have  thus  the 
theorem  : 

III.  The  integral     \  f(z]dz,  taken   along   the   boundary   of  a 
domain  in  which  the  function  is  regular  except  at  a  finite  number 


FIG.  21 


238  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

of  poles,  is  equal  to  2  -rri  times  the  sum  of  the  residues  of  the  function 
at  these  poles. 

Up  to  this  time  we  have  tacitly  assumed  that  the  domain 
under  consideration  excluded  entirely  the  point  at  infinity ;  in 
considering  also  domains  which  contain  the  point  oo  as  an 
inner  point,  we  must  first  determine  what  is  to  be  understood 
by  the  residue  of  a  function  at  the  point  oo  .  We  must  therefore 
observe  that  when  z  is  replaced  by  z'  (cf.  §  21),  dz  is  replaced 

by  —  z'  dz' ;  the  integral  I  f(z)dz  is  then  replaced  by 
—  I  z'  <f>(z')dz' ;  and  this  integral  taken  along  a  closed  curve  is 

then  zero  if  z'  </>(V),  that  is,  z1  '/(z)  is  regular  inside  of  this  curve. 
The  fundamental  theorem  of  §  35  must,  therefore,  be  modified 
as  follows  for  a  domain  which  contains  the  point  oo  within  it : 

IV.  The  integral   \f(z)dz,  taken   along  a   dosed  curve   which 

incloses  the  point  oo ,  is  equal  to  zero  if  z*  'f(z)  is  regular  inside  of 
the  domain  inclosed  by  this  curve  and  containing  the  point  oo  . 

But  if  s2  •  f(z)  is  not  regular  inside  of  this  curve,  it  follows  in 
consideration  of  the  integral  \  f(z)dz  — —  i(f>(z')'z'  •  dz' 
that: 

V.  In  the  application  of  Theorems  II and  III  to  a  domain  which 
contains  the  point  oo  within  it,  we  are  to  take  as  the  residue  at  this 
.point  the  coefficient  of  z~l  *  with  the  opposite  sign  in  the  develop 
ment  (III,  §  44). 

Every  closed  curve  on  a  sphere  divides  it  into  two  parts  and 
can  be  looked  upon  as  the  boundary  of  each  of  these  parts.  If 
a  function  be  regular  in  each  of  these  parts  except  for  particular 
poles — which  is  only  the  case  for  rational  functions  according 

*  Not 


§45-    CAUCHY'S  THEOREM   ON   RESIDUES  239 

to  VI,  §  44  —  we  can  apply  The 
orem  III  to  each  of  the  parts. 
The  same  integral  appears  both 
times,  but  the  direction  of  inte 
gration  is  opposite  (that  is,  taken 
each  time  so  that  the  domain  con 
sidered  lies  to  the  left).  If  we 
now  add  the  two  results,  the  inte 
grals  disappear,  and  we  have  the 
following  theorem : 

VI.    The  sum  of  all  the  residues 
of  a  rational  function  is  always  equal  to  zero. 

EXAMPLES 

1.  Consider  the  function         *        .    What  are  its  poles  ?    Find 
its  residues  at  the  points  —  i,  —  <o,  —  or.  (  =  -§-,  f  w,  -J  w2,  respec 
tively.)     Find  the  value  of  J  —^ — -  dz  taken  along  a  semicircle 

and  its  diameter,  having  its  center  at  the  origin  and  including 
the  two  points    —  <u,  —  or  (to  is  a  primitive  3d  root  of  unity). 

Ans.    —  -f  7i7". 

2.  If  in  the  CAUCHY  formula  /(a)  =  — ^-   C*^'      ,  z  —  a  is 

2  TriJ^z  —  a) 

replaced  by  (z  —  a^  •  (z  —  a2)  •••  (z  —  #„),  prove  that 

1  r_        /  (zXg v  /(^A)  .    j    ^ 

2  7riJr(z  -  a^(z  -  a2)  •••  (z  -  av)      ^/'  (a^    (z  —  a^' 

3.  If  f(z)  has  a  simple  zero  z  =  a  but  no  pole  in  the  finite 
domain  S  bounded  by  C,  then 

f(z)dz^ 


240  IV.    SINGLE-VALUED   ANALYTIC  FUNCTIONS 

4.    Show  that  the  residue  of 


(z-a)(x-z) 
at  the  point  z  =  a  is  . 

Show  also  that  the  residue  of  the  function 

A 


at  this  same  point  is 


(x  -  a) 


5.    Determine  the  residue,  also  the  logarithmic  residue  of  the 
function 


(z-a](z-b^ 
at  the  point  z  =  b. 

§  46.   The  Theorem  concerning  the  Number  of  Zeros  and  of  Poles. 
A  Second  Proof  of  the  Fundamental  Theorem  of  Algebra 

If  f(z)  is  a  function  of  a  complex  argument  z  a.ndf'(z)  its  first 

f'(,,\ 

derivative,  we  shall  call  J-  -M  the  logarithmic  derivative  of  f(z) ; 

f(z) 

the  reason  for  this  nomenclature  will  appear  later.  Other 
theorems  concerning  the  function  f(z)  may  be  obtained  if  we 
use  the  logarithmic  derivative  instead  of  f(z)  itself  in  the  appli 
cation  of  the  theorems  of  the  above  paragraph  ;  for  this  purpose 
a  few  theorems  concerning  the  logarithmic  derivative  are 
needed  : 

I.    If  f(z)  is  regular  in  the  neighborhood  of  the  point  ZQ  and  dif 
ferent  from  zero  at  that  point,  then  <-±-}  is  regular  there. 


§  46.    SECOND  PROOF  OF  FUNDAMENTAL  THEOREM        241 

For,  in  that  case,   -^—  (according  to  X,  §  38)  and  also  f\z) 
(according  to  VII,  §  38)  are  regular  in  the  neighborhood  of  z0. 
II.    If  f(z)  is  regular  in  the  neighborhood  of  a  point  ZQ  and  has 

an  m-fold  zero  at  that  point,  then  *  ^  '  has  a  simple  pole  at  ZQ  and 
its  residue  there  is  m. 

For  then  we  can  put  (A.  A.  §  24) 

(0  /(*)  =  (*-%)-•/(«) 

where  f\(z)  is  understood  to  be  a  function  regular  in  the  neigh 
borhood  of  ZQ  and  different  from  zero  at  z0  ;  it  follows  from  this 
that: 


f(z)       «-«,/i(«) 

Since  "_LJ  is  regular  in  the  neighborhood  of  ZG  according  to 

f\(z} 

I,  the  correctness  of  Theorem  II  is  evident  from  this  equation. 
It  is  proven  in  the  same  way  that  : 

III.    If  f(z]  has  an  m-fold  pole  at  z  =  %  *  ^  '    has  a  simple 

f(z] 
pole  with  the  residue  —  m  at  that  point 

The  proof  of  Theorems  I  to  III  is  understood  to  be  valid  for 
a  point  situated  in  the  finite  part  of  the  plane.  For  the  point 
at  infinity  Theorem  I  is  unchanged,  but  in  Theorems  II  and  III 
only  the  statements  relating  ti  the  values  of  the  residues  remain 

true,  not  the  statement  that  £-&  has  a  simple  pole.     On  the 

/(*) 

contrary,  the  logarithmic  derivative  is  regular  at  infinity  even  in 
these  cases.  This  is  however  irrelevant  in  the  application  which 
we  now  wish  to  make  since  we  are  concerned  only  with  the 
residues. 


242  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 


By  applying  Theorem  III,  §  45,  to  ~~  we  obtain  the  follow 
ing  theorem  : 

IV.    The  integral          — 


2 


taken  in  the  positive  sense  along  the  boundary  of  a  domain  in  which 
the  function  f(z)  is  everywhere  regular  except  at  poles,  is  equal  to 
the  number  of  zeros  of  f  (z)  in  this  domain  diminished  by  the  num 
ber  of  poles ;  every  zero  and  every  pole  is  to  be  counted  here  as  often 
as  its  order  of  multiplicity  indicates. 

Further,  we  find  from  Theorem  VI  of  §  45  that : 

V.  Every  rational  function  becomes  zero  as  often  as  infinite  upon 
the  sphere  (which  is  only  another  formulation  of  Theorem  III, 
§21); 

and  if  we  apply  it  to/(z)  —  c  instead  of/(0),  we  find  that : 

VI.  A  rational  function  takes  on  any  arbitrary  vahie  c  just  as 
often  as  it  becomes  infinite. 

In  these  theorems  too,  multiple  zeros  or  poles  are  to  be 
counted  according  to  their  order  of  multiplicity ;  the  expression 
"f(z)  takes  on  the  value  f(z)  =  c  n  times  at  the  point  z  =  a,"  means 
that  c  is  the  first  term  in  the  development  of  f(z]  in  powers  of 
z  —  a,  for  which  terms  with  i,  2,  •  •  •,  (n—  i)st  powers  of  (z  —  a) 
do  not  appear,  but  the  term  (z  —  d]n  is  present. 

In  particular,  a  rational  integral  function  of  the  nth  degree  is 
everywhere  regular  except  at  infinity  and  has  an  ^-fold  pole  at 
infinity  ;  it  therefore  follows  from  Theorem  V  that : 

VII.  Every  rational  integral  function  of  the  nth  degree  has  n 
zeros;  or,  expressed  otherwise: 

VIII.  Every  algebraic  equation  of  the  nth  degree  has  n  roots. 
We  thus  have  a  second  proof  of  the  fundamental  theorem  of 

algebra  (cf.  VII,  §  44). 


§  46.    THE  NUMBER   OF   ZEROS  AND   OF   POLES         243 

It  follows  further  from  this  that  a  rational  fractional  function 
has  as  many  poles  as  its  degree  indicates  (II,  §  20).  For,  if  the 
degree  m  of  the  numerator  is  not  greater  than  the  degree  n  of 
the  denominator,  its  degree  is  n  ;  it  is  then  regular  at  infinity 
and  has  ;/  poles  in  the  finite  part  of  the  plane.  But  if  ;;/  >  n, 
its  degree  is  equal  to  ;;/  and  it  has  an  (;//  —  ;j)-fold  pole  at  infin 
ity  in  addition  to  the  ;/  poles  in  the  finite  part  of  the  plane. 
From  Theorem  VI  it  thus  follows  that  : 

IX.  Every  rational  function  takes  on  any  arbitrary  complex 
value  as  of  fen  as  its  degree  indicates. 

We  make  further  use  of  Theorem  IV  in  order  to  deduce  an 
important  extension  of  Theorem  VIII  of  §  38.  Let  w  =f(z]  be 
a  function  regular  in  a  circle  about  the  origin  and  /'(o)  =£  o  ; 
without  loss  of  generality,  we  giay  assume  that  w  =  o  for  z  =  o, 
since  this  can  always  be  obtained  by  a  parallel  translation  of 
the  w-plane.  We  can  then  take  r  so  small,  according  to  VIII, 
§  39,  that  no  other  zeros  of  f(z)  lie  inside  or  upon  the  circum 
ference  of  a  circle  F  of  radius  r,  and  thus,  according  to  IV  : 


(3) 


If  therefore  ;;/  be  the  smallest  value  which  |  f(z)  \  assumes  on  F, 
and  Wi  any  value  of  w  whose  absolute  value  is  smaller  than  m, 
then  the  number  of  roots  which  the  equation 

/CO  =  «'i 

has  inside  of  F  is  : 

(4)  n=-^C     ™        >dz. 

2  Tri  Jr  /(O  -  «>i 

If  we  put 

(5)  I--^)  =  A 


244  IV-    SINGLE-VALUED   ANALYTIC   FUNCTIONS 


it  follows  that  ^(z)  =  wi  ' 

[/(*)]' 


- 


Equations  (3)  and  (4)  therefore  give  : 


the  last  integral  taken  along  that  curve  C  of  the  /-plane  which 
corresponds  by  means  of  equation  (5)  to  the  circle  T  of  the 
s-plane.  But  since  \f(z)  \  ^  m  >  \  wv  on  T  according  to  the 
hypothesis,  C  can  never  be  so  far  from  the  point  /  —  i  that  it 
could  inclose  the  point  t  =  o;  thus  integral  is  then  zero,  n  =  i, 
and  the  equation  f(z]  =  u\  has  one  and  only  one  root  inside  of 
T.  But  it  follows  from  VIII,  §  39,  that  we  can  also  describe  in 
the  z-plane  a  circle  about  the  origin  of  so  small  a  radius  p  (<  r) 
that  \f(z]  in  it  takes  on  only  values  which  are  <  m.  The 
theorem  is  therefore  as  follows  : 

X.  If  '  w  =f(z)  is  a  function  of  z  regular  in  the  neighborhood  of 
the  point  z  =  o  and  f\o]  3=  o,  a  circle  of  so  small  a  radius  can 
then  be  drawn  about  this  point  that  w  takes  on  different  values  at 
different  points  in  it,  and  that  (  VIII,  §  38)  the  values  which  w 
takes  on  in  this  circle  cover  a  region  U  of  the  w-plane  inside  of 
which,  conversely,  z  is  a  regular  function  of  w. 

For  the  actual  construction  of  this  function  in  individual  cases, 
we  can  make  use  of  the  method  of  undetermined  coefficients 
(A.  A.  §  78,  §  79)  ;  or  we  could  use  a  theorem,  also  useful  other 

wise,  obtained  by  applying  Theorem  III  of  §  45   to   z  '  *  ^- 

/W 

This  function  is  regular  where  /(s)  is  regular  and  different  from 


§  46.    SECOND  PROOF  OF  FUNDAMENTAL  THEOREM      245 

zero;  at  an  m-fold  zero  a  oif(z)  it  has  the  residue  ma,  at  an 
w-fold  pole  b,  the  residue  mb.  (Both  are  true  for  a  =  o,  b  =  o, 
respectively  but  not  for  a=  oo  ,  or  £  =  oo  .)  Therefore,  we 
obtain  : 

XL    The  integral       — .  f 

2   7T/  J 

&&;;  /»  the  positive  se'nse  along  the  boundary  of  a  domain  lying  in 
the  finite  part  of  the  plane  in  which  /(z)  is  everywhere  regular  ex 
cept  at  poles,  is  equal  to  the  sum  of  the  zeros  of f(z)  in  this  domain 
diminished  by  the  sum  of  the  poles,  multiple  zeros  or  poles  being 
counted  as  often  as  their  orders  of  multiplicity  indicate. 

If,  instead  of  using  the  function  /(z)  itself,  we  apply  this 
theorem  to  the  function  /(z)  —  w  and  to  the  domain  in  which 
this  function  has  only  one  zero  and  no  poles,  we  obtain  the  fol 
lowing  theorem : 

XII.    The  integral    —,   f  4y4^  dz> 

2  TTlJ     f(Z)  —  W 

taken  in  the  positive  sense  along  the  circle  defined  in  Theorem  X, 
represents  the  solution  of  the  equation 


for/^\  and  in  fact  that  solution  which  belongs  to  the  region  U  de 
fined  in  Theorem  X. 

If  we  expand  here  under  the  integral  sign  in  increasing 
powers  of  w  and  then  integrate  term  by  term,  we  obtain  for  this 
solution  a  development  in  powers  of  w  which  converges  inside 
of  the  largest  circle  that  can  be  drawn  about  the  zero  point  of 
the  ov-plane  and  which  belongs  entirely  to  the  region  U. 

In  connection  with  these  conclusions  we  discuss  another 
theorem  which  appeared  earlier  in  a  different  discussion  of  this 

theory.     The  integral  /• 

|  udv 


246  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

taken  along  the  boundary  of  a  domain  in  the  z/zvplane,  represents, 
as  will  be  taken  for  granted  here,  the  area  of  this  domain  ;  and 
it  has  the  positive  or  the  negative  sign  according  as  the  bound 
ary  is  described  in  the  positive  or  in  the  negative  sense  in  the 
process  of  integration.  If  we  introduce  x  and  y  as  variables  of 
integration  in  this  integral,  regarding  u  and  v  as  functions  of  x 
and  y,  we  obtain  the  integral : 


taken  along  the  corresponding  curve  of  the  jf_y-plane.  If  this 
curve  incloses  a  domain  whose  map  upon  the  corresponding 
domain  of  the  z^-plane  is  reversibly  unique,  then  the  value  of 
the  integral  is  positive  when  taken  around  the  domain  of  the 
jrj'-plane  in  the  positive  sense,  and  negative  in  the  opposite  case 
if  the  sense  of  the  angle  remains  unchanged  throughout  the 
mapping.  The  first  is  always  the  case  according  to  the  last 
theorems  if  u  +  iv  is  an  analytic  function  of  x  -f-  iy  and  the 
domain  is  sufficiently  small.  But  since  an  integral  taken  over 
an  arbitrary  curve  can  always  be  replaced  as  in  §  29  by  a  sum 
of  integrals  over  sufficiently  small  curves,  it  follows  that : 

XIII.    If  u  +  iv  is  a  regular  function  of  x  +  iy  over  the  whole 
domain  inclosed  by  a  curve  F,  then  the  integral 


taken  in  the  positive  sense  along  F,  is  always  positive. 

The  only  exception  to  this  theorem  occurs  when  the  function 
u  +  iv  maps  the  domain  under  consideration  in  the  x -\- iy-p\ane 
not  in  general  upon  a  domain,  but  upon  a  single  point,  that  is, 
when  it  is  constant.  (The  conceivable  case  of  mapping  the 
domain  of  the  x  -f  /y  plane  upon  a  curve  of  the  u  +  iv  -plane  is 


§47-    THE   LAURENT'S   SERIES  247 

not  possible  on  account  of  the  Theorems  V,  §  26  ;  VIII,  §  38  ; 
X,  §  46.)  To  include  this  exception  in  the  formulation  of 
Theorem  XIII,  we  must  say  "  never  negative  and  only  zero 
when  //  +  iv  is  constant "  instead  of  "  positive." 

By  means  of  this  theorem  we  may  obtain  a  second  proof  of 
the  fundamental  Theorem  IV,  §  44.  Theorem  XIII  is  also 
valid  for  a  part  of  the  sphere  which  includes  the  point  oo  as  an 
inner  point,  provided  that  the  function  //  +  iv  is  regular  in  this 
domain  in  the  sense  of  definition  I  of  §  44.  However,  we  must 
in  this  case  take  for  positive  direction  of  integration  that  one 
for  which  the  domain  under  consideration,  as  also  the  point  at 
infinity,  lies  to  the  left. 

If  now  we  have  a  function  which  is  regular  over  the  whole 
sphere,  we  can  divide  the  sphere  into  two  parts  by  any  curve 
which  does  not  go  through  the  point  infinity,  and  we  can  then 
apply  Theorem  XIII  to  each  of  these  two  parts.  It  then  fol 
lows  first,  that  the  integral  cannot  be  negative  when  we  take 
the  part  lying  on  the  finite  part  of  the  sphere  always  to  the  left ; 
and  second,  that  it  cannot  be  negative  when  the  part  containing 
infinity  lies  to  the  left.  These  two  conditions  are  together  pos 
sible  only  when  the  integral  is  zero.  But  then  the  function 
u  +  iv  is  constant,  Q.  E.  D. 

§  47.     The  LAURENT'S  Series 

In  §  36  we  studied  CAUCHY'S  theorem  for  a  domain  5  which 
had  one  bounding  curve.  We  return  now  to  this  theorem,  study 
ing  it  for  a  domain  S  in  which  the  function  /(z)  is  known  to  be 
regular  and  which  has  two  bounding  curves  T,  y  (cf.  Fig.  16). 
Equation  (3)  of  §  36  also  holds  in  this  case  ;  but  the  integration 
is  performed  along  each  of  the  curves  T,  y  in  such  direction 
that  the  domain  S  lies  to  the  left.  To  evaluate  this  integral  in 
the  positive  sense  along  each  of  the  two  curves,  we  must  change 


248  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

the  sign  of  the  integral  taken  along  y  ;  the  given  equation  then 
takes  the  form  : 


Let  us  study  in  particular  the  case  for  which  T,  y  are  concen 
tric  circles  about  the  origin,  and  S  the  annular  domain  bounded 
by  these  two  circles.  Then,  since  £  is  understood  to  be  a 
point  within  S,  |  £  |  <  |  z  |  for  all  elements  of  the  first  integral, 
which  can  therefore  be  developed,  just  as  in  §  37,  in  powers  of  £ 
with  increasing,  positive,  integral  exponents.  But  for  all  ele 
ments  of  the  second  integral  and  for  all  points  £  within  S 


accordingly,  the  series 


i       z 


*-{        £     £2  £" 

converges  uniformly  for  all  such  pairs  of  values  (z,  £).  It  may 
therefore  be  integrated  term  by  term,  and  a  development  is  ob 
tained  for/(£)  of  the  form  : 


(3)  +  tf-iS-1  +  a.  2£~2  +  -  •  -  +  <*_„£-  +  •  •  • 

the  coefficients  of  which  are  expressed  by  integrals  as  follows  : 

/  \  i     Cf(z)dz 

(4)  *»  =  —  .  I  y-~r>  «  =  °>  i,  2  ..-, 


(5)  *-  =  —      ^-1  -A*)^,  n  =  i,  2,  3  •  •  , 

2   7T/4/Y 

The  two  formulas  (4)  and  (5)  can  be  combined  into  one  by 
making  use  of  Theorem  VII,  §  35.  In  consequence  of  this 
theorem  T  and  also  y  can  be  replaced  by  any  curve  C  lying  in 


§47-    THE   LAURENT'S  SERIES 


249 


this  ring,  which  has  the  property  that  it  divides  the  ring  into 
two  parts,  each  of  which  is  annular  (one  part  bounded  by  T  and 
C,  the  other  by  C  and  y).  We  then  state  the  resulting  theorem 
as  follows,  making  use  of  a  generally  accepted  notation  for  writ 
ing  series  of  the  form  (3) : 

I.  If  a  function  f(z)  is  regular  in  a  ring  bounded  by  two  circles 
concentric  about  the  origin,  it  can  then  be  developed  inside  of  this 
annular  domain  in  a  convergent  series  of  the  form: 


(6) 


which  contains  an  infinite  number  of  terms  in  powers  of  £  with 
positive  as  well  as  negative  exponents.  The  coefficients  of  this 
series  are  expressed  by  the  integrals : 


(7) 


2  TTl 


dz, 


taken  along  any  curve  C  which  surrounds  the  origin  once  and  lies 
entirely  inside  the  circular  ring. 

Such  a  series  is  called  a 
LAURENT'S  Series. 

Particular  attention  should 
be  given  the  cases  where  the 
radius  of  F  is  increased  indefin 
itely  or  that  of  y  decreased 
indefinitely,  providing  the  func 
tion  always  satisfies  the  con 
ditions  of  the  theorem  in  the 
domain  thus  enlarged.  Both 
cases  appear  at  the  same  time 

if  we  consider  a  function  which  is  regular  over  the  whole  sphere 
with  the  single  exceptions  of  the  points  z  ==•  o  and  s  =  oo. 


250  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

Conversely,  let  us  suppose  that  for  a  function  /"(£)  a  develop 
ment  of  the  form  (3)  is  found  which  converges  inside  of  the 
annular  domain  between  two  circles  T,  y;  and  in  fact,  to  fix 
this  hypothesis  more  precisely,  let  each  of  the  two  series 

(8) 

be  convergent  inside  of  the  annular  domain.  Then  the  first 
series,  according  to  III,  §  38,  converges  uniformly  in  every 
domain  which  lies  entirely  inside  of  F,  the  second  converges 
uniformly  in  every  domain  which  lies  entirely  outside  of  y. 
Hence  both  series  converge  uniformly  on  a  curve  such  as  C  in 
Fig.  23,  and  hence  they  may  be  integrated  term  by  term  along 
this  curve.  •  Let  us  do  this  after  first  multiplying  by  £~m~1 ;  then, 
in  connection  with  equations  (10)  and  (n)  of  §35,  we  find: 

(9) 

and  this  coincides  with  (7) ;  that  is,  therefore, 

II.  When  a  function  can  be  developed  in  a  series  of  the  form  (j) 
which  converges  in  the  given  sense  inside  of  the  circular  ring  be 
tween  F  and  y,  then  the  coefficients  have  the  values  given  by  (/) ; 
this  development  is  therefore  unique. 

The  last  statement  requires  some  explanation  in  order  that  it 
may  have  only  the  intended  meaning.  A  function  may  be 
regular  inside  of  different  circular  rings,  e.g.,  between  yl  and  y2 
between  y2  and  y3,  while  upon  y2  there  are,  for  example,  poles  of 
the  function.  Theorem  I  is  then  applicable  to  each  of  these  two 
rings  and  two  LAURENT'S  expansions  are  thus  obtained,  one  of 
which  converges  between  yl  and  y2  and  the  other  between  y2 
and  y3 ;  and  we  are,  therefore,  not  to  understand  Theorem  II  to 


§47-    THE   LAURENT'S   SERIES 

mean  that  these  two  expansions  must  have  the  same  coefficients. 
On  the  contrary,  Theorem  II  is  applicable  only  to  the  expansion 
inside  of  one  and  the  same  ring. 

Thus,  for  example,  we  obtain  for  the  expansion  of 


Z2  —  3  -S  +  2         Z  —  2         Z  —   I 

inside  of  the  circle  of  unit  radius  about  the  point  z=  o : 
between  this  circle  and  the  circle  of  radius  2  : 


outside  of  the  latter  :      -f  -1  +  3_  +  1  -f  •  •  •  . 

z1      zr      z 

The  generalization  of  the  theorems  of  this  paragraph  to  the 
case  where  the  two  concentric  circles  have  not  the  point  z  =  o 
but  any  other  arbitrary  point  as  center  is  treated  as  in  VI,  §  39, 
and  requires  no  further  explanation. 

EXAMPLES 

1.  Develop —  in  a  series  of  integral  powers  of  z 

2-3      z-i 

valid  for  the  domain  in  which  this  function  is  regular. 

2.  Expand  — - —  inside  a  circle  whose  center  is  O ;  that  is, 

i  —  z 

expand  in  powers  of  z.     How  large  may  the  circle  of  conver 
gence  be  ? 

3.  Expand  -  inside  a  circle  whose  center  is  the  point  / ;  that 
is,  in  powers  of  z  —  i. 


252  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

4.  Expand  1  inside  a  circle  whose  center  is  the  point  —  i  ; 

z1 
that  is,  in  powers  of  z  +  1. 

5.  Expand  —  inside  a  circle  whose  center  is  the  point  —  /  ; 
that  is,  in  powers  of  z  +  i- 

6  Expand  ---  I  --  in  powers  of  z.     Locate  the  circle  of  con- 

O+i)3 
vergence. 

7  Expand  -  -  -  in  powers  of  (z  —  i).     Locate  the  circle 

O  +  i)3 

of  convergence. 

8.    Given  the  function  f(z)  =  T^  —  .     It  has  singular  points 
at  z  _  ±  j  because  e*  is  holomorphic*  over  the  whole  finite  part 

of  the  plane.     Let  us  put  -£-  =  -&-  +  -&-  +  H(z)  where 

z2  —  i      2—i      0+1 

jy(0)  denotes  a  function  holomorphic  over  the  whole  finite  part 

of  the  plane  ;  in  other  words,  it  is  a  function  for  which  the  dis 

continuities  of  the  original  function  are  removed.     It  is  required  to 

I.    Determine  Cv  and  C2  and  H(z)  ; 

II.    Expand  H(z)  for  a  circle  whose  center  is  the  point  z=o; 

III.  Expand  H(z)  for  a  circle  whose  center  is  the  point  z=  i  ; 

IV.  Expand  H  (z)  for  a  circle   whose  center   is   the   point 

f  SB—   I. 


=  (2  _  i)-i  .  {a  series  in  powers  of  (z  —  i)} 


2" 


*  The  term  holomorphic  is  used  in  reference  to  such  functions  which  are  JMf^- 
valued,  regular  (tnonogenic}  ,  and  continuous  in  the  given  domain.  —  S.  E.  R. 


§  47-    THE   LAURENT'S   SERIES 


253 


is  a  function  having  no  singularity  at  z  =  +  I. 


Thus 


In  the  same  way  eliminate  also  the  singularity  at  z  =  —  I  and  find  £3. 


Hence 


— ^ ^-  must  be  H(z). 

*  -  I         •  -  1         2+1 


9.   A  function  having  poles  of  order  one  at  alt  of  order  two  at 
#2,  and  of  order  three  at  #3,  etc.,  in  a  domain  A,  is  generally  of 
the  type 
r*          f*  r  c1  C*  f* 

Ll        -  C2  |  3  |  L*  [  U5  j  U6          |    , 


where  -^T(s)  is  a  function  holomorphic  in  the  domain  A  contain 
ing  01?  <72i  «3?  •••!  an^  is  thus  a  function  for  which  the  discontinu 
ities  of  the  original  function  are  removed.  The  constants  C\, 
etc.,  may  be  found  as  in  the  previous  example. 


10. 


(s  +  i)(*+ 4) 

I.    Inside  a  circle  whose  center  is  the  point  o ; 
II.    Inside  a  circle  whose  center  is  the  point  oo  ; 

III.  Inside  a  circle  whose  center  is  the  point  2  • 

IV.  In  the  circular  ring  which  excludes  the  points  —  i ,  —  4. 
HINT.  —  To  expand  /(z)   in 

2+1         2+4 

For  — - — ,  B  is  in  the  domain 

2+  I 

at  infinity  since  it  is  outside  of 
the  domain  in  which  is 


2  + 


regular ;    hence 


is    ex 


panded  in  powers  of  —     For 

z 

— - — ,B  is  in  the   domain  in 


254  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

which  it  is  regular;  hence  expand  for  a  circle  with  center  at  o,  that  is,  in 
powers  of  z.  Substitute  these  in  f(z)  and  the  expansion  for  the  domain  B 
is  obtained. 

11.  Suppose  f(z)  and  <j>(z)  have  at  the  point  z  =  a  poles  of 
order  m  and  n  respectively.     What  can  be  said  of  the  behavior 
of  the  functions  //^ 

/(*)  -  $(z\  /(*)  +  *(*),  -US 

4>(z) 

at  this  point  ?     Discuss  all  cases. 

12.  Suppose  f(z)  has  an  ;«-fold  zero  at  z  =  a.     Show  that  the 
integral 


has  an  (m  -f-  i)-fold  zero  there. 

State  the  analogous  proposition  for  the  integral 


in  the  neighborhood  of  a  pole  a. 

§  48.    Behavior  of  a  Regular  Function  in  the  Neighborhood  of 
a  Critical  Point 

We  may  frequently  prove  that  a  function  is  in  general  regular  in 
a  domain,  but  the  proof  may  fail  for  particular  points  of  this 
domain,  so  that  the  question  as  to  the  behavior  of  the  function 
at  these  critical  points  remains  undetermined.  A  certain 
amount  of  information  is  furnished  in  such  cases  by  the 
LAURENT'S  series. 

Let  the  origin  be  such  a  point,  that  is,  let  the  function  /(z)  to 
be  investigated  be  regular  at  every  point  of  a  certain  neighbor 
hood  of  the  origin  with  the  exception  of  the  origin  itself,  con 
cerning  which  nothing  is  known.  The  circle  y  used  in  connec 
tion  with  LAURENT'S  theorem  can  then  be  taken  arbitrarily 
small. 


§48.    A   REGULAR   FUNCTION  NEAR   A   CRITICAL   POINT     255 

And  when  |  f(z]  \  always  remains  less  than  an  assignable  limit 
however  near  z  may  approach  the  origin,  it  follows  that  the  coeffi 
cients  a_n  (5,  §  47)  must  all  be  equal  to  zero.  But  then  the 
LAURENT'S  expansion  of  f(z)  represents  a  function  regular  at 
the  origin ;  and  if  removable  discontinuities  be  excluded  as 
agreed  upon  in  §  43,  it  follows  that  this  function  must  coincide 
with/(s)  even  at  the  origin.  Hence  the  following  theorem  : 

I.  When  a  function  of  a  complex  argument  is  regular  in  the 
neighborhood  of  the  origin,  this  point  itself  excepted,  and  when, 
in  arbitrarily  approaching  the  origin,  it  remains  in  absolute 
value  always  less  than  any  assignable  limit,  then  the  function  is 
regular  at  the  origin  itself  provided  that  removable  discontinuities 
are  excluded. 

This  may  be  expressed  more  briefly  but  less  exactly  as  follows  : 
"  A  function  of  a  complex  argument  is  everywhere  continuous 
where  it  is  finite." 

But  if  in  the  LAURENT'S  expansion  of  the  function  in  the 
neighborhood  of  the  point  z  =  o  terms  with  negative  exponents 
appear,  we  must  determine  whether  there  are  an  infinite  or  only 
a  finite  number  of  such  terms.  In  the  first  case  the  function 
behaves  at  the  point  z  =  o  just  as  a  transcendental  integral 
function  at  infinity  (X,  §  44)  ;  that  is,  it  approaches  arbitrarily 
near  to  every  value  in  every  neighborhood  of  this  point.  For, 
the  sum  of  the  terms  with  positive  exponents  becomes  arbitrarily 
small  in  a  sufficiently  small  neighborhood  of  the  point  z=o  and 
it  is  only  a  question  of  the  terms  with  negative  exponents.  In 
the  second  case  the  function  is  definitely  infinite  at  z  =  o  in  the 
following  sense : 

When  a  positive  number  M  howmer  large  is  given,  we  can  always 
draw  a  circle  about  the  point  z  =  o  with  a  radius  sufficiently  small 
{but  >  o)  so  that  \  f(z)  \  >  M  for  all  points  inside  of  it.  But,  if  in 


256  IV.    SINGLE- VALUED   ANALYTIC   FUNCTIONS 

this  second  case  the  pole  is  ;z-fold,  the  limit 
(i)  limi(s  -«)"•/(«)! 

z±a 

exists  and  is  finite  and  different  from  zero ;  on  the  contrary  for 
every  positive  c  (however  small) 

lim{(*-*)"+«.X*)}=o 

z±a 

and  lim  \(z  —  a)n~e  -X2)!  *s  definitely  infinite. 

z=a 

By  means  of  the  following  general  definition  : 

II.  A  function  is  said  to  &?  definitely  infinite  and  of  the  ^th 
order  at  z  =  a  when  the  limit 

Hm  i(*-«f/fti 

z±a 

exists  and  is  finite  and  different  from  zero  —  we  may  state  the 
theorem : 

III.  When  a  function  of  a  complex  argument  is  regular  and 
single-valued  in  the  neighborhood  of   a  point  a,   the  point  itself 
excepted,  and  becomes  definitely  infinite  at  a,  it  is  always  infinite 
at  a  of  an  assignable  integral  order. 

EXAMPLES 


1.  Expand  v(f  —  d](z  —  b)  in  powers  of  z  for  the  neighbor 
hood  of  the  point  z  =  oo   (cf.  equations  8,  9,  §  62). 

2.  Write  the  power  series  which  represents  the  function  /(z) 
in  the  neighborhood  of  the  point  z  =  oo, 

i st.    If  the  point  z  =  oo  is  an  ordinary  point  for  the  function  ; 

2d.    If  the  point  z  =  oo  is  a  zero  of  order  m  ; 

3d.    If  the  point  z  =  oo  is  a  pole  of  order  m  for  the  function. 


§  49-    FOURIER'S  SERIES 


257 


3.    Expand 


O-i) 


each  in  the  neighborhood  of  the  point  z  =  oc  . 

4.    Expand   the   functions  of   the   previous   example   in   the 
neighborhood  of  the  point  2=3. 


§  49.    FOURIER'S  Series 

From  the  LAURENT'S  expansion  valid  in  a  ring  between  two 
concentric  circles,  we  can  derive  an  expansion  valid  in  a  strip 
bounded  by  two  parallel  lines  by  making  use  of  §§  40-42.  Let 
r  and  R  be  the  radii  of  the  two  circles  between  which  a  function 
f(z)  satisfies  the  conditions  of  LAURENT'S  theorem.  Without 
loss  of  generality,  we  may  suppose  r<  i  and  R  >  i  ;  when  this 
is  not  originally  the  case  it  can  be  obtained  by  introducing  cz  in 
place  of  z  where  c  is  understood  to  be  a  suitably  chosen  real 
constant.  We  can  then  put : 

(1)  r=e~mi,    R  =  emz 

where  mlt  m2  denote  positive  real  constants. 
By  means  of  the  function 

(2)  ,  =  * 


FIG.  24 


we  can  then  map  a  rectangle  of  the  /-plane  upon  the  circular  ring 
of  the  2-plane ;  we  are  to  think  of  this  ring  as  having  a  cut  (or 


258  IV.    SINGLE-VALUED   ANALYTIC  FUNCTIONS 

slit)  along  the  negative  real  axis;  and  if  we  put  t=-t^-\-  ttf, 
the  equations  of  the  sides  of  this  rectangle  become  : 

(3)  *!  =  —*,   /i  =  +  TT,   /2  =  —  ^2,    /2  =  mv 

Now  —  exists  and  is  finite  and  different  from  zero  everywhere 


inside  of  this  rectangle,  and  therefore  /(«)=  <KO  is  a  regular 
function  of  /;  and  the  LAURENT'S  series  : 

/(«>=  §<•„*•, 

becomes  :  +„ 


or,  by  introducing  the  trigonometric  instead  of  the  exponential 
functions  : 


(5)        <KO  •»  *  +(*.  +  O  cos  **f  *fe  -  '-J  sin  «'• 

w=l  n=l 

Conversely,  if  a  function  of  /  is  regular  inside  of  the  rectangle, 
it  is  transformed  by  substitution  (2)  into  a  function  of  z  regular 
inside  of  the  circular  ring  opened  along  the  cut.  But  in  the 
application  of  LAURENT'S  theorem  it  is  necessary  that  f(z)  be 
regular  inside  of  a  circular  ring  not  opened  along  the  cut.  This 
is  the  case  when,  and  only  when,  <£(/)  is  also  regular  at  least  in 
narrow  strips  beyond  the  sides  of  the  rectangle  parallel  to  the 
/2-axis  and  besides  when  <£(/)  takes  on  the  same  values  at  pairs  of 
points  on  these  sides  which  have  the  same  coordinates  /2.  For 
then  the  transference  of  the  neighborhood  of  these  two  sides  to 
the  2-plane  gives  two  functions  of  z  regular  in  the  neighborhood 
of  the  cut,  which  coincide  along  the  cut,  and  are  therefore  in 
general  identical  according  to  I,  §  39.  In  particular*  this  is 

*  When  a  function  satisfies  the  foregoing  conditions,  we  can  always  look  upon 
it  as  a  piece  of  a  periodic  function  regular  in  the  parallel  strip. 


§  49-    FOURIER'S   SERIES  259 

the  case  when  the  function  <£(/)  is  periodic  with  period  2  TT  and 
is  regular  in  the  entire  parallel  strip  bounded  by  the  straight  lines 
/2  =  —  w2  and  /2  =  m\-  We  can  therefore  state  the  following 
theorem  : 

I.  A  periodic  function  with  the  period  2  TT  which  is  regular  in  a 
strip  having  a  finite  breadth  along  both  sides  of  the  real  axis,  can 
be  developed  in  a  series  (a  "  FOURIER'S  Series  ")  of  the  form  : 

30  QC 

<£(/)  =  aQ  +  ^  an  cos  ;//  -f  ^  bn  sin  nt 

«=1  n=l 

which  is  uniformly  convergent  and  admits  term  by  term  derivatives 
of  all  higher  orders. 

The  coefficients  of  this  series  are  determined  by  introducing  / 
as  variable  of  integration  by  means  of  substitution  (2)  in  the 
representation  of  the  coefficients  of  the  LAURENT'S  series  : 

'.  =  —  :(/>)*—  '* 

2   TTlJ 

given  in  7,  §  47.     It  is  thus  found  that 


(6)  <*n  =  fn  +  '-»  =  ~    f#M  COS  *****  (H   > 

irJ 

t>n  =  i(*n  ~  t-n)    =  L     f^W  ^  "*  <&, 

•nJ 


where  these  integrals  are  to  be  taken  along  any  curve  which 
connects  a  point  of  the  side  4  =  —  TT  with  the  point  lying  oppo 
site  on  the  side  ^  =  TT,  the  simplest  way  of  connecting  them 
oeing  then  to  use  the  real  values  between  /  =  —  TT  and  /  =  +  TT. 


260  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

§  50.    Sums  of  an  Infinite  Number  of  Regular  Functions 
Let 

(i)  /!«,/««,  •••/.«••• 

be  an  infinite  sequence  of  functions  of  z  which  are  all  regular 
inside  of  a  definite  domain  B  of  the  2-plane  ;  let  it  be  assumed 
further  that  the  series 


converges  at  every  point  of  this  domain.  Its  sum  is  then  a 
complex  function  (I,  §  31)  of  the  real  coordinates  x  and  y  of 
z  —  x  +  iy.  Nothing  more  can  be  asserted  concerning  this  sum 
unless  we  make  additional  assumptions  concerning  the  func 
tions  fn(z). 

But  if  we  assume  further  that  series  (2)  converges  uniformly 
in  the  entire  domain  under  consideration,  we  can  show  as  fol 
lows  that  its  sum  represents  a  function  F(z)  of  a  complex  argu 
ment  z  regular  inside  of  this  domain.  If  F  is  the  bounding 
curve  of  this  domain,  we  may  integrate  series  (2)  term  by  term 
along  this  curve,  since  according  to  hypothesis  it  converges 
uniformly  along  F.  Moreover,  this  remains  true  if  we  divide  by 
z  —  £  before  integrating,  providing  the  denominator  does  not 
become  indefinitely  small  at  any  point  on  the  path  of  integra 
tion  ;  this  provision  is  satisfied  when  £  is  an  inner  point  of  the 
domain  (not  a  point  on  the  boundary).  If  therefore  the  sum  of 
series  (2)  be  designated  provisionally  by  S(z),  we  obtain  : 


2  ir         z- 


When  the  origin  belongs  to  the  domain  and  £  is  nearer  to  it 
than  all  of  the  boundary  points,  we  can  expand  in  increasing 
powers  of  £  under  the  integral  sign  on  the  left-hand  side  of  this 


§50.    SUMS   OF  AN   INFINITE  NUMBER   OF  FUNCTIONS     261 

equation  and  integrate  term  by  term  as  in  §  37  ;  we  thus  see 
that  this  left-hand  side  is  in  the  neighborhood  of  the  origin  a  reg 
ular  function,  F(£),  of  £,  from  which,  to  be  sure,  we  cannot  con 
clude  merely  on  the  basis  of  this  representation  by  integrals  that 
it  is  identical  with  .S(£).  However,  the  right-hand  side  is 


since  the  separate  functions  /„  are  by  hypothesis  regular  in  B. 
Hence  S(£)  =  J?(£).  Since  the  origin  can  be  put  at  an  ar 
bitrary  point  of  the  domain  we  state  the  theorem  : 

I.  The  sum  of  a  series  of  regular  functions  uniformly  convergent 
in  a  connected  domain  is  itself  a  regular  function  inside  of  this 
do  mam. 

This  theorem  follows  also  from  XIII,  §  38  :  if  the  integrals  of 
the  separate  terms  are  equal  to  zero  for  every  closed  path  of 
integration  inside  of  B,  the  same  is  true  for  the  integral  of  the 
sum  on  account  of  the  uniform  convergence. 

A  sum  of  the  form  (2)  may  under  certain  conditions  be  uni 
formly  convergent  in  each  of  several  domains  not  connected 
among  themselves.  It  will  then  represent  a  regular  function  in 
each  of  these  domains  ;  but  this  does  not  warrant  the  conclu 
sion  that  these  functions  are  connected  with  each  other  in  any 
manner  whatever.  As  a  matter  of  fact,  simple  examples  *  show 
that  such  connection  does  not  necessarily  exist. 

Moreover,  we  can  draw  further  conclusions  from  equation  (3). 
If  a  be  any  point  inside  of  the  domain  S,  we  obtain  : 


dz 


*  Cf.,  for   example,   WEIERSTRASS,  Ges.    Werke,  Vol.  II,  pp.  213,  231.     Also 
FORSYTHE,  Theory  of  Functions,  p.  138.  — S.  E.R. 


262  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

and  therefore,  according  to  (8),  §  39  : 


On  account  of  the  uniforrh  convergence  of  the  series  we  may 
here  interchange  summation  and  integration,  and  thus  obtain  : 


that  is,  the  following  theorem  is  true  : 

II.  A   series  of  regular  functions,   uniformly  convergent  in  a 
definite  domain  (not  merely  along  a  curve],  admits  term  by  term 
derivatives  of  all  higher  orders  inside  of  its  domain  of  convergence. 

From  this  theorem  it  then  follows  further  that  : 

III.  In  order  to  obtain,  according  to  TAYLOR'S  theorem,  the  ex 
pansion  of  a  regular  function  which  is  defined  by  a  series  of  regular 
ftmctions  uniformly  convergent  in  the  neighborhood  of  z  =  a,  we  may 
expand  each  term  of  the  series  in  powers  of  z  —  a  and  then  collect  all 
terms  having  the  same  powers  of  z  —  a. 

§  51.   MiTTAG-LEFFLER'S  Theorem 

A  function  F(z)  which  is  everywhere  regular  over  the  finite 
part  of  the  plane  except  for  a  finite  number  of  poles  a^a^  •••  ,  an, 
can  always  be  represented,  according  to  VI,  §  44,  in  the  form  : 

(0  %/•&+&)> 

v=i 

in  which  g(z)  represents  a  transcendental  integral  function  of  z 
and  fv(z)  a  rational  function  having  no  poles  other  than  av. 
Closely  allied  to  this  is  the  investigation  of  the  question 


§  51.    MITTAG-LEFFLER'S  THEOREM  263 

whether  a  function  with  an  infinite  number  of  poles  can  also  be 
represented  in  the  form  of  an  infinite  series  of  partial  fractions : 

w 

For  this  purpose  it  is  necessary  and,  according  to  the  results  of 
§  50,  also  sufficient  that  the  series  be  uniformly  convergent. 
We  can  see  from  simple  examples  that  such  is  not  always  the 
case  when  the  poles  av  and  the  functions  /v(z),  which  .determine 
how  F(z)  becomes  infinite,  are  arbitrarily  prescribed.  The 
difficulty  arising  in  this  way  was  surmounted  by  MITTAG- 
LEFFLER  by  demonstrating  that  rational  integral  functions  gv(z) 
can  always  be  so  determined  that  the  series : 

(3) 

converges  uniformly.  We  shall  not  give  here  a  proof  for  the 
most  general  case,  but  concern  ourselves  only  writh  a  generaliza 
tion  *  sufficing  for  most  applications. 

It  is  to  be  noticed  in  the  first  place  that  when  the  function  is 
regular  everywhere  over  the  finite  part  of  the  plane  except  at 
poles,  then  the  set  of  points  av  cannot  have  a  limit  point  in 
the  finite  part  of  the  plane  (IV,  §  43).  Therefore  an  infinite 
number  of  the  points  av  cannot  lie  in  a  finite  domain  (XVI, 
§  25) ;  in  other  words,  we  must  have 

(4)  lim  \av    =  oo  . 

We  shall  now  first,  suppose  that  |  av  increases  so  rapidly  as  v 
increases  that  an  integer  n  can  be  determined  which  has  the  prop 
erty  that  the  series : 

(5) 


*  For  the  general  case  cf.  WEIERSTRASS,  Ges.  Werke,  Vol.  II,  p.  189. 


264  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

converges.  Second,  let  us  suppose  that  the  above  poles  are  all 
of  the  same  order  A  and  that  the  decomposition  into  partial 
fractions  of  each  of  the  functions  fn  consists  of  only  one  term 
with  coinciding  coefficients,  all  of  which  may  then  be  taken 
equal  to  i  ;  let  then 

(6)  /M  *(*-**)-*> 

First,  let  A  =  n  :  let  any  finite  domain  be  given  which  con 
tains  none  of  the  points  av ;  let  M  be  the  largest  value  which 
\z  takes  in  this  domain,  yu,  any  positive  number  greater  than  i. 
We  then  divide  the  points  av  into  two  classes  according  as 
I  ^  I  =  P^f or  I  av  \  >  P-M.  According  to  hypothesis  there  are 
only  a  finite  number  of  the  points  of  the  first  class,  say  k ;  for 
every  point  av  of  the  second  class  and  for  every  point  z  of  the 
given  domain 

(7) 


z—  a. 


-i 
< 


that  is,  smaller  than  a  finite  number  independent  of  z  and  v.    Let 
us  now  subtract  the  finite  sum : 


»(*-«,)" 

from   the   series   to  be  investigated  and  there  will  remain  the 
infinite  series 

i 


Each  term  of  this  series  arises  from  the  corresponding  term  of 
series  (5)  by  multiplication  by  the  «th  power  of  the  factor  (7), 
from  wrhich  it  appears  that  for  all  terms  it  is  less  than  one  and 
the  same  finite  limit.  Since  series  (5)  according  to  hypothesis 
converges  absolutely,  series  (9)  also  converges  absolutely 
(A.  A.  §  56);  and,  in  fact,  converges  uniformly  since  the  above- 


§51.    MITTAG-LEFFLER'S   THEOREM  265 

mentioned  limit  is  independent  of  z.     Let  us  again  add  the  first 
terms  (8)  and  thus  obtain  the  theorem  : 

I.    If  the  series  (5)  converges,  then  the  series 


converges  absolutely  and  uniformly  in  every  domain  which  lies 
in  the  finite  part  of  the  plane  and  which  contains  none  of  the 
points  av. 

Second:  if  A.  >  n,  then  the  terms  of  the  series  : 


arise  from  the  corresponding  terms  of  series  (10)  by  multipli 
cation  by  the  factors  : 

(12)  (*-<O-x+*. 

If  small  circles  of  radius  p  be  described  about  the  points  av  and 
if  z  be  limited  to  a  domain  containing  none  of  these  circles, 
then  each  of  the  factors  (12)  for  all  points  z  of  this  domain  is  in 
absolute  value  less  than  the  finite  number 


independent  of  z  and  v.     But  since  series  (10)  converges  abso 
lutely,  it  follows  that  : 

II.  If  series  (5)  converges,  then  for  \  >  n  series  (n)  also  con 
verges  uniformly  and  absolutely  in  every  domain  which  lies  in  the 
finite  part  of  the  plane  and  which  contains  none  of  the  points  av. 

But  third:  if  X  <  n,  —  X  +  n  is  positive  and  we  cannot  draw 
the  conclusion  as  above  ;  for  then  \z  —  av\  ~x+n  is  not  smaller 
but  is  larger  than  p~A4n.  But  in  this  case  we  may  proceed  as 
follows  :  By  integrating  term  by  term  between  two  arbitrary 


266  IV.    SINGLE-  VALUED   ANALYTIC   FUNCTIONS 

limits  z0,  z  along  an  arbitrary  path  inside  the  domain  of 
uniform  convergence  —  which  is  allowable  according  to  VIII, 
§  28  —  we  obtain  the  following  uniformly  convergent  series 
from  series  (10)  : 


In  this  series  each  term  becomes  infinite  for  z  =  at,  as  -  -  - 

0  -  a,)1-1 

does  ;  the  problem  is  thus  solved  for  A  =  n  —  i.  We  can  apply 
the  same  conclusion  to  this  series  when  n  >  2,  and  so  continue 
until  the  exponents  in  the  denominator  are  depressed  to  X.  We 
state  this  result  explicitly  only  under  the  simple  supposition 
that  the  point  z  =  o  is  not  one  of  the  points  av  ;  we  may  then 
put  zQ  =  o  and  thus  obtain  the  following  theorem  : 

III.    If  series  (5)  converges  and  if\<.n,  then  the  series 


converges  uniformly  and  absolutely  in  every  domain  which  lies  in 
the  finite  part  of  the  plane  and  which  contains  none  of  the  points 
av  in  its  interior,  provided  that  the  whole  expression  under  the 
summation  sign  be  considered  as  one  term  of  the  series  and  is  not 
separated. 

Moreover,  the  law  of  formation  for  series  (14)  can  be  stated 
thus  :  To  (z—a^)~K  must  be  added  a  rational  integral  function  of 
z  such  that  every  term  of  the  series  is  zero  of  order  n  —  \  at 
the  point  2  =  0. 

It  therefore  follows  from  the  general  theorem  of  §  50  that 
each  of  the  series  (10),  (n),  (14)  represents  a  function  of  z 


§  Si.    MITTAG-LEFFLER'S  THEOREM  267 

regular  in  the  domain  of.  its  uniform  convergence.  Its  behavior 
at  one  of  the  points  av  may  be  obtained  by  taking  out  of  the 
series  that  term  which  is  relevant  at  this  point  ;  the  remaining 
part  of  the  series  also  converges  uniformly  in  the  neighborhood 
of  av,  it  is  then  regular  there,  and  the  function  has  therefore  a 
pole  at  z  =  av  of  the  kind  prescribed. 

IV.  The  most  general  function,  which  has  poles  of  the  prescribed 
kind  at  all  these  points,  is  obtained  by  adding  the  most  general 
transcendental  integral  function  to  the  sum  of  the  series. 

If  it  is  a  question  of  developing  a  preassigned  function  in  a 
series  of  partial  fractions  of  the  kind  here  considered,  the 
determination  of  this  complementary  integral  function  presents 
a  certain  difficulty  which  can  be  disposed  of  in  some  cases  by 
the  following  procedure  due  to  CAUCHY. 

Let  us  suppose  an  infinite  sequence  of  closed  lines  Cv 
(v  =  i,  2,  3,  •••)  having  the  property  that  each  time  Cv_^  lies 
entirely  inside  of  Cv  and  the  point  av  lies  inside  of  Cv  but  out 
side  of  £"„_!.  If,  therefore,  small  circles  are  drawn  about  the 
points  «!,  a2,  ~*,ak,  Theorem  II,  of  §  45  is  applicable  to  the 
domain  between  Ck  and  these  circles;  we  obtain  accordingly: 

/«)  = 


2  TT 


The  problem  is  then  solved  if  by  any  suitable  choice  of  the 
curves  Ck  we  succeed  in  determining  the  value  of  the  limit  to 
which  the  integral  standing  on  the  right  converges  as  k  =  oo.* 
In  the  application  to  individual  cases  this  method  may  be  modi 
fied  in  various  ways  ;  for  example,  inside  each  of  the  curves  Cv 
we  may  take  two  poles  more  than  in  the  preceding  instead  of  one. 

*  For  further  investigations  cf.  E.  PlCARD,    Traite  d'  analyse,  Vol.  II    (Paris, 
1893),  chap.  VI,  No.  5  et  seq. 


268  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

§  62.    Decomposition  of  Singly  Periodic  Functions  into  Partial 

Fractions 

We  return  now  to  the  investigation  of  singly  periodic  func 
tions  discontinued  at  §  42  having  in  the  meantime  obtained 
additional  methods.  The  theorem  of  the  last  paragraph  enables 
us  to  form  a  priori  such  functions. 

By  introducing  cz  instead  of  z  as  the  argument,  i  can  be  made 
the  primitive  period  for  the  function.  It  is  required  to  form  a 
function  having  this  period  and  having  a  pole  at  z  =  o ;  it  must 
then  necessarily  have  poles  at  all  those  points  which  arise  from 
the  point  z  =  o  by  the  addition  and  subtraction  of  periods  ;  that 
is,  in  the  points  : 

z  =  i,  2,  3,  •••  ,   co  ,    z  =—  i,  —  2,  —  3,  ...  ,  co. 

Let  us  form  now  a  function  which  has  these  points  (and  no 
others)  for  poles  ;  and,  in  order  to  apply  the  theorems  of  the 
previous  paragraph,  we  inquire  whether  there  is  any  value  of 
n  for  which  the  series 

^A  i 


converges.  The  reader  will  readily  recall  that  this  series  is  not 
convergent  for  n  =  i,but  does  converge  for  n  —  2  (A.  A.  §  55). 
There  is  then  according  to  (10),  §  51,  a  function : 


(O 


which  has  all  of  the  points  named  above  for  a  twofold  pole. 
In  forming  from  it,  according  to  (14),  §  51,  another  function  for 
which  these  points  are  only  simple  poles,  we  observe  that  the 
hypothesis  made  there  does  not  apply  here,  viz.  that  the  point 
z  =  o  is  not  to  be  among  the  points  av.  Therefore,  in  applying 


§  52.  PARTIAL  FRACTIONS  OF  SINGLY  PERIODIC  FUNCTIONS  269 


that  theorem  here,  we  must  do  so  not  toyi(z)  but  to/^z)  — 
the  following  function  is  thus  obtained  : 


w 


(The  accent  on  the  summation  sign  signifies  here  and  in  what 
follows  that  the  value  v  =  o  is  omitted  from  the  values  over 
which  the  summation  is  taken.) 

We  can  now  show  that  these  two  functions  constructed  in 
this  way  really  have  i  for  a  period.  That  the  first  function  has 
the  period  i  follows  directly  from  the  representation  (i).  For, 
if  we  replace  z  in  this  representation  by  z  +  i ,  we  obtain  in 
full  the  following : 


i  -  v)2      (z  +  i)2       A  (s  +  i  -  v)5 


Replacing  now  the  summation    letter  v  in  this  expression  by 
fji  +  i ,  we  obtain  : 


and  this  is  the  original  series,  except  that  the  term  s~2  is  com 
bined  with  the  first  sum  and  the  term  (z  -f-  i)~2  is  omitted  from 
the  second  sum.  It  therefore  follows  that : 

(3)  /.(«+0  =•/.(»)• 

But  the  same  conclusion  for  the  function  /2(2)  cannot  be  drawn 
since  the  parenthesis  in  (2)  is  not  to  be  removed  :  however, 
since 

(4)  /,(»)= - 


270  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

it  follows  from  equation  (3)  by  integration  that 

(5)  /2(*  +  i)=/2«+c; 

In  this  equation  C  signifies  a  constant  of  integration  which  can 
be  determined  whenever  we  evaluate  both  sides  of  the  equa 
tion  for  some  particular  value  of  z.  We  could  also  use  a  value 
of  z  for  which  the  two  sides  become  infinite  ;  for  this  purpose 
we  compare  the  first  terms  of  the  expansions  valid  for  the 
neighborhood  of  this  value  of  z.  Thus,  for  example,  we  obtain 
for  the  neighborhood  of  z  =  o  : 


_!_  +  !=_£+(#), 

z  —  v      v  v 


and  therefore  : 
(6) 

also: 


z  -f-  i 


Z  +  I  —  I 


2-f-I  —  V        V         I   —  V        V         (t   —  v)5 

and  accordingly 


Comparison  of  the  coefficients  of  z°  gives 

00  —00 

r—~_l_\-\       l        I  ^ 

—         i      /    "~7  \     '      /      /  \  * 

z,v(!_v)    ^Ki-y) 

If  v  be  replaced  by  i  —  /x  in  the  second  summation,  we  obtain : 


§52.  PARTIAL  FRACTIONS  OF  SINGLY  PERIODIC  FUNCTIONS  2/1 

The  two  summations  are  therefore  equal  to  each  other,  and  in 
fact  each  is  equal  to  —  i  ;  for, 


-i     .m  m 


Accordingly  C  =  o  ;  that  is  : 

I.  Not  only  fi(z)  but  also  f-z(z)  is  a  periodic  function  of  z  with 
tJte  period  I. 

We  inquire  next  about  the  relation  of  these  functions  to  the 
periodic  functions  investigated  in  §§  40-42  ;  to  answer  this 
question,  we  make  use  of  the  method  due  to  CAUCHY  mentioned 
at  the  end  of  the  previous  paragraph.  We  observe  that  in 
equation  (2)  the  terms  may  be  arranged  in  pairs  of  values  of  v 
that  are  equal  but  opposite  in  sign  ;  accordingly  then 


(where  as  before  the  parenthesis  must  not  be  removed).  If  the 
cotangent  function  is  now  denned  as  for  real  variables  by  the 
equation  : 

(9)  cot,  =  <?*», 

sin  z 

it  follows  from  the  results  of  §  41  that  the  function  TT  cot  (TTZ) 
has  the  same  (simple)  poles  and  the  same  residues  as  f*(z).  If 
we  then  -take  as  the  line  Ck  a  rectangle  whose  sides  have  the 

equations:  „  k 

x  =  ±  ti^L!  and  _>'  =  ±  77, 

the  poles:  o,  ±  i,   ±  2,  •••,  ±  k 

lie  inside  of  it  ;  we  therefore  obtain  : 

(,o)      .cot  K)  =1  +         _!_  +  -JU-L, 


2/2 
But: 


IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 


7T  COt  (  TTZ 


since  the  cotangent  is  an  odd  *  function  and  the  line  Ck  is  sym 
metrical  about  the  point  2=0;  we  can  then  replace  the  integral 
appearing  in  (10)  by 


To  evaluate  this  integral  let  us  start  from  the  equation : 

(12)  |«>t(«)  ,_J"^-'"^      --- 


£ny     I     ,,—2ny  


2  cos  2 


it  follows  from  this  that,  upon  the  two  vertical  sides  of  the  rec 
tangle  : 

(13)  |  cot  (**)!  = 


and  upon  the  two  horizontal  sides : 


(14)        |  COt  (IT*)  |  ^ 


;  that  is,  rg 


i  +  e 


Therefore,  along  all  of  the  lines  Ck,  |  cot  (irz)  <  Hf,  where  M 
denotes  a  number  independent  of  k.  If  we  designate  the  short 
est  distances  of  the  points  o  and  £  from  Ck  by  rk  and  pk,  and  the 
length  of  Ck  by  <5^,  it  then  follows  that : 


f CQt  (* 

Jc,  z(z  - 


dz 


< 


M-S 


*  An  odd  function  may  be  defined  as  one  for  which  f( x)  =  — f( —  x)  and  an  even 
function  one  for  which  f(x)  —  +/(—  x) .  Particular  cases  are  those  for  which  the 
odd  function  contains  only  odd  powers  of  the  variable  and  an  even  one  only  even 
powers,  as  sin  x  and  cos  x.  —  S.  E.  R. 


§  53-     THEOREMS  ON  SINGLY  PERIODIC  FUNCTIONS         273 

As  rj  increases,  M  decreases  ;  we  are  thus  at  liberty  to  allow 
rj  to  increase  to  infinity  with  k.  It  follows,  therefore,  that 
Sk  =  8  rk  and  ph  (for  a  given  £)  increases  to  infinity  with  k. 
Then  the  limit  of  the  integral  as  k  =  oo  is  equal  to  o  and  it  fol 
lows  from  (10)  that  : 

(l6)  fo(?)   =  TTCOt  TTZ 

(cf.  A.  A.  (i  i),  §  84)  and  from  this  it  follows  further  that  : 


II.  The  functions  represented  by  these  partial  fractions  are  then 
rational  functions  of  cos  irz  and  sin  TTZ. 

If  in  equations  (16)  and  (2),  a  and  a  +  z  are  substituted  suc 
cessively  for  z  and  the  results  subtracted,  the  following  formula 
is  obtained  : 

(18)      ,[cot  ,(a  +  z)-  «*(«,)]  = 


§  63.     General  Theorems  concerning  Singly  Periodic   Functions 

We  derive  here  another  general  theorem  concerning  singly 
periodic  functions  for  which  Theorem  II  of  the  previous  para 
graph  is  a  special  case.  Let  us  again  suppose  that  i  is  the  primi 
tive  period,  since  it  may  be  obtained  by  multiplying  the  argument 
by  a  constant,  and  that  then  a  strip  bounded  by  the  lines  x=  —  ^ 
and  x=  -f  %  can  be  used  as  the  period  strip  ;  and  we  study  singly 
periodic  functions  f(z),  which  have  the  following  properties  : 

1.  f(z]  is  everywhere  regular  in  the  finite  part  of  the  plane 
except  at  poles. 

2.  When   z  =  x-\-ty  passes   to   infinity  where  y  is   positive 
without  going  outside  of  the  period  strip,  at  least  one  of  the  two 

limits  \\rnf (z)  or  lim  (  -^— ]  exists. 

„-+»        »*+«v/(*/ 


2/4  IV-    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

3.  When  z  =  x  -f-  iy  goes  to  infinity  in  the  same  manner 
where  y  is  negative,  we  have  analogous  results  ;  however,  it  is 
not  supposed  that  \im/(z)  =  \imf(z). 

y=-co  y=  +  °o 

By  means  of  the  substitution  : 

(1)  £  =  <**• 

we  can  map  (cf.  §§42  and  49)  the  parallel  strip  of  the  s-plane 
conformally  on  the  ^-sphere  cut  along  a  meridian  (apart  from 
the  neighborhoods  of  the  points  £  =  o  and  £  =  oo).  In  this  way 
the  function  /(  z)  is  transformed  into  a  function  <£(£)  which  has 
the  following  properties  : 

1.  Since  f(z)  is  periodic,  <£(£)  is  single-valued;  its  values  (as 
also  those  of  its  derivative)  on  one  side  of  the  cut  pass  continu 
ously  into  the  values  on  the  other  side  of  the  cut. 

2.  Since  f(z)  is  regular  everywhere  in  the  finite  part  of  the 
plane    except  at  poles,   <£(£)  is  regular,  with  the  exception  of 
poles,  over  the  whole  sphere  except  at  £  =  o  and  £  =  oo. 

3.  If  £  is  allowed  to  converge  to  zero  along  any  path,  then 
the  corresponding  s-path  runs  to  infinity  where  y  is  positive  ; 
and  if  the  £-path  does  not  encircle    the  point  £  =  o  infinitely 
often,  then  the  s-path  first  crosses  a  finite  number  of   period 
strips  and  finally  remains  within  one  of  them.     It  follows  there 
fore  from  hypothesis  (2)  that  at  least  one  of  the  two  limits 

(2)  li 


exists  (for  every  such  kind  of  approach  of  £  to  zero),  and  that 
then  (I,  III,  §  48)  <£(£)  is  either  regular  at  £  =  o  or  has  a  pole 
there.  But  even  when  the  £-path  does  encircle  the  point  £  =  o 
an  infinite  number  of  times,  the  s-path  crosses  an  infinite  num 
ber  of  period  strips  and  we  obtain  the  same  result  ;  for,  since 
f(z)  is  supposed  to  be  periodic,  we  can  transfer  to  the  first  strip 
all  parts  of  the  3-path  which  lie  in  strips  other  than  the  first  one. 


§  53-     THEOREMS   ON   SINGLY   PERIODIC   FUNCTIONS          2/5 

4.  In  an  analogous  manner,  it  follows  from  hypothesis  (3) 
that  <£(£)  is  either  regular  at  infinity  or  has  a  pole  there. 

Thus  <£(£)  is  regular  over  the  whole  sphere  except  at  poles, 
and  is  therefore,  according  to  VI,  §  44,  a  rational  function  of  £ ; 
that  is,  we  have  the  theorem  : 

I.  Every  periodic  function  which  satisfies  the  hypottieses  (/)-(j), 
is  a  rational  function  of  the  expotiential  function  e2"*2, 

A  series  of  further  theorems  follow  from  this  one.  Let  f(z) 
be  such  a  function  ;  with  the  aid  of  equation  (n),  §  40,  we  can 
then  eliminate  the  exponential  functions  from  the  expressions 
for  f(zi),  f(z*),  f(zi  +  &>)  formed  according  to  Theorem  I,  and 
obtain  an  algebraic  equation  between  /(^  +  &),  /(%),  /(^2), 
whose  coefficients  are  independent  of  z^  and  z2.  Such  an  equa 
tion  is  called  an  algebraic  addition  theorem  ;  hence  the  theorem: 

II.  Every  function  of  the  kind  described  has  an  algebraic  addi 
tion  tlworem. 

Further,  if  we  had  two  such  functions,  we  could  eliminate  the 
exponential  function  and  find  that : 

III.  Between  pairs  of  such  functions  there  is  an  algebraic  equa 
tion  with  coefficients  independent  of  z. 

In  particular  this  is  true  of  such  a  function  and  its  first 
derivative;  accordingly,  we  have: 

IV.  Every  such  function  satisfies  an  algebraic  differential  equa 
tion  of  the  first  order  in  which  the  independent  variable  does  not 
appear  explicitly  ; 

or  otherwise  expressed : 

I V  a.  Every  such  function  is  the  inverse  of  the  integral  of  an 
algebraic  function. 


2/6  IV.    SINGLE-  VALUED   ANALYTIC   FUNCTIONS 

The   following   equations,    for  real  values  of  u  and   z,    are 
examples  of  this  theorem  : 


(4) 


Theorem  III  introduces  us  to  a  class  of  algebraic  equations 
between  two  variables  z  and  j,  which  are  satisfied  identically  by 
putting  single-valued  singly  periodic  functions  of  an  auxiliary 
variable  u  (a  "  uniformizing  *  variable  "  one  may  say)  equal  to 
s  and  z\  as,  for  example,  in  the  equation  : 


where 

(6)  s  —  sin  u,    z  =  cos  u. 

But  we  recall  that  these  equations  (on  account  of  Theorem  I) 
are  none  other  than  those  which  are  satisfied  identically  by  put 
ting  rational  functions  of  an  auxiliary  variable  equal  to  z  and  j, 
for  example,  in  equation  (5)  : 

(7)  *  =  ^ 


We  will  not  introduce  here  the  proof  that  this  property  does 
not  belong  to  every  algebraic  equation  between  two  vari 
ables.  On  the  contrary,  the  investigation  of  single-valued 
functions  of  a  complex  variable  is  discontinued  at  this  point 
and  the  discussion  of  many-valued  functions  is  taken  up  in 
the  next  chapter. 

*  Cf.  WHITTAKER,  Modern  Analysis,  p.  338.  — S.E.R. 


MISCELLANEOUS   EXAMPLES 


MISCELLANEOUS    EXAMPLES 

1.  Determine  all  the  roots  of  the  equations : 

(a)  sin  z  =  2,    (&)  cos  z  =  —  5  /'. 

2.  Show  that  the  functions  sin  z,  cos  z,  have  no  other  zeros 
or  no  other  periods  than  those  of  the  real  functions  sin  x,  cos  x. 

3.  If  C(.v)  =  i  -  —  +  —-...  and  S(x)  =#—  —  +——  ... 

2!      4!  3!      5! 

show  that       C(x  +  y)  =  C(x)  •  C(y)  -  S(x)  •  S(y) 
and  S(x  +v)  =  S(x)  •  C(y)  -f  C(x)  -  S(y). 

4.  Prove  that  a  function  which  has  a  derivative  that  vanishes 
at  every  point  of  a  finite  region  is  constant  in  that  region. 

5.  How  is  a  definite  integral  defined  for  a  complex  variable  ? 
From  the  definition,  show  that 

t)as  I  <  ML 

where  L  is  the  length  of  the  path  of  integration  and  M  is  the 
maximum  of  \f(z)  \  on  this  path. 


6.    State  and  prove  CAUCHY'S  theorem  on  residues. 

X+a 
, 


.,2  y 


f   I 
(.v2 


Cf.  also  Ex.  35  of  this  list  and  the  reference  given  there. 

7  a.    An  integral  appearing  in  the  theory  of  probability  is  the 
following  one  :  ^. 

I    <r*dx. 

Jo 


2/8  IV.    SINGLE-VALUED   ANALYTIC   FUNCTIONS 

The  following  method  of  evaluation  (cf.  PICARD,  Traite  (T  analyse, 
Vol.  i,  p.  104)  is  particularly  simple  and  elegant.  Let  us  begin 
with  the  real  double  integral : 


taken  over  the  first  quadrant.     This  integral  converges    since 
the  following  limit  exists  : 

lim  rke-**-y\    ;2  =  x*  +  /,    k>  2. 

X=ao ,  y=x, 

We  now  obtain  the  desired  formula  by  putting  the  double  inte 
gral  in  the  form  : 


and  evaluate  it  by  means  of  polar  coordinates  in  the  form 

S.T- 

Thus 


2 


8.  By   taking     I  e'^dz    along   the    rectangle    y  =  o,  y  =  a, 

x  =  ±  /?,  prove  that 

J      e~x*  •  cos  2  ax  •  dx  —  VTT  •  e~<* 

/»+» 
given  that  I      e-J  .  ax=  VTT. 

*s  —<n 

gae 

9.  What  are  the  poles  of  the  function  -  ? 


10.  Prove  that  any  two  simply  connected  plane  regions  can 
be  mapped  conformally  on  each  other,  stating  accurately  the 
theorems  used  in  the  proof. 


MISCELLANEOUS   EXAMPLES  2/9 

11.    If  the  functions  ft(z)  can  be  developed  about  the  point 

z  =  o  as  follows  :     ... 


and  if  the  series  : 


is    uniformly  convergent   throughout  the  neighborhood  of  the 
point  z  =  o,  prove  that  the  series  : 

£0  =  aQ  +  A)  +  ^  +  •  •  • 

£1  =  #i  +  ^i  +  A  -h  •  •  • 

^2  =  #  2  +  ^2  +  ^2  + 

are  convergent  and  that  the  development  of  P(z)  is 
/r(s)«4+4f  +  4*+  •••. 

12.  Establish  the  relation  between  the  convergence  of  a  series 
of  complex  terms  and  the  convergence  of   the  series  of   their 
absolute  values. 

13.  If  a  polynomial  in  (x,  y)  with  real  coefficients  satisfies 
LAPLACE'S  equation,  prove  that  it  is  the  real  part  of  a  polyno 
mial  in  z  —  x  +  iy. 

14.  Prove   that  a  necessary  and   sufficient  condition  that  a 
homogeneous  polynomial  of  the  nth  degree  in  (x,  y)  satisfies 
LAPLACE'S  equation  is,  that  the  equation  formed  by  setting  the 
polynomial  equal  to  zero  represents  n  real  straight  lines  making 

angles  —  with  one  another. 
n 

15.  Define  the  exponential  function  for  complex  values  of  the 
argument,  pointing  out  the  chief  characteristics  which  must  be 
preserved  in  order  that  the  new  function  may  be  regarded  as  a 
generalization  of  the  original  one. 

16.  Prove  that  a   rational  function   can    be    represented  by 
means  of  partial  fractions. 


280  IV.    SINGLE-  VALUED   ANALYTIC   FUNCTIONS 

17,  What  functional   properties  characterize   completely  the 
rational  functions  ?     (Cf.  also  Ex.  6  at  the  end  of  §  68.) 

18.  Discuss  the  theory  of  the  system  of  partial  differential 
equations  . 


where  P  and   Q  are  given  functions  of  x  and  y,  and  show  how 
this  system  of  equations  is  connected  with  the  theory  of  the  line 

integral  /• 

J  (Pdx  +  Qdy). 

What  connection  has  this  with  functions  of  a  complex  variable  ? 

19.  What  is  the  condition  that   a  function  /(z),  having  the 

2rn 

period  <o,  be  expressible  as  a  rational  function  e™  ? 

20.  Define  the    terms  :    Singular    Point  ;    Pole  ;    Order  of  a 
Pole;  Critical  Point;  Order  of  a  Critical  Point. 

State  and  prove  the  geometrical  property  of  a  critical  point  of 
the  nth  order. 

21.  Define  GREEN'S  theorem  for  two  dimensions  and  explain 
its  physical  meaning.     (Cf.  HARKNESS  AND  MORLEY,  Introduc 
tion,  etc.,  p.  322.) 

Show  how  GREEN'S  theorem  may  be  applied  to  a  simply  con 
nected  region  to  effect  the  conformal  mapping  of  the  region  on 
the  interior  of  a  circle. 

22.  Give  a  direct  proof  that  in  the  transformation  by  means 
of  w  =  ez  angles  are  preserved. 

23.  Suppose  a  function  holomorphic  in  a  region  A  with  the 
exception  of  poles  at^,  c*,  •••1cp  of  order,  respectively,  nlt  n2,  •  •,  np. 
Discuss  the  general  type  of  the  function  which  is  holomorphic 
everywhere  in  the  region  A  ;  that  is,  find  the  function  for  which 


MISCELLANEOUS   EXAMPLES  28 1 

the  discontinuities  of  the  original  function  are  removed.     (Cf. 
Exs.  8,  9,  at  the  end  of  §  47.) 

24.  Prove  that  for  sufficiently  large  values  of    |  z  \  ,  the  abso 
lute  value  of  the  last  term  in 

aTzr  +  ar+lzr+1  +  "-  +anzn 

where  r  is  an  integer  which  is  less  than  n  and  greater  than  o,  is 
greater  than  the  sum  of  the  absolute  values  of  the  remaining  terms. 

25.  Prove  that  the   sum  of  two  functions,   both   continuous 
at  a,  is  continuous  at  a.     Prove  the  same  for  their  product,  and 
also  for  their  quotient  if  the  denominator  is  not  zero. 

26.  Calculate  the  residues  of  the  function    —   *       +1>    and 
then  show  that 

dx          _  i.  3.  5.  •••  (2  n  —  i) 
'-oo  (i  +_T2)n+1  ~~  2.4.6.  •••        2  n 

and  derive  from  the  latter  result  the  value  of  the  integrals 


£ 


* and   <**  Jx 


C)' 

irou 
region  bounded  by  the  curve  along  which  the  integral  is  taken.     In  this  case 


HINT.—  /nO)  =—  f  fl*}  '  dz    ,  where  /(s)  is  regular  throughout  the 

2  7TZ  J    (2  —   a)n  +  1 


27.  If  /(z)  is  single-valued  and  regular  in  a  region  S,  show 
that  i/f(z)  is  in  general  regular  in  this  region.      Discuss  the 
singularities  of  the  latter  function. 

28.  When  is  a  function  f(z)  said 

(a)  to  be  "  analytic  about  "  or  "regular  at  "*  the  point  z  =  00, 
(ft)  to  have  a  root, 

(c)  to  have  a  pole, 

,  7N  A    ,  , .  ,      at  the  point  z  =  co  ? 

(d7)  to  have  an  essential 

singular  point 

*  Cf.  BOCHER,  Bull.  Am.  Math.  Soc.,  Vol.  Ill,  p.  89.  — S.  E.  R. 


282  IV.    SINGLE-VALUED   ANALYTIC  FUNCTIONS 

29.    If  the  f unctions  /(z),  <f>(z)  each  have  an  essential  singular 
point  at  the  point  z  =  oo,  what  can  be  said  about  the  function 


Give  the  reasons  for  your  answer. 

30.  If  F(z)  and  G(z)  are  rational  integral  functions  of  z  of 
degree  «  and/  respectively,  show  directly  that  there  is  one  and 
only  one  pair  of  rational  integral  functions  of  z  : 

Q(z)  =  q^-v  +  ft*-*-1  -f  '  •  •  ?»_„ 
G,(z}  =  r^-1  +  rlST*  +  ...  r^ 
which  satisfies  the  identity 

F=QG  +  6V 

Develop  the  right  side  in  powers  of  z  and  equate  coefficients  of 
like  powers  on  both  sides  of  the  equation.  This  gives  ;*  +  i 
linear  equations  for  the  determination  of  the  «  +  i  coefficients  : 

^0>   *  *  *}  ^n-pl      rQ>  "  *>  rp-l- 

31.  A  single-valued  function  w  =  f(z)  of  z  is  called  periodic 
when  there  is  a  constant/  =£  o  such  that/(s  +  /)=/(s)  for  every 
value  of  z.       Show  directly  from   the  fundamental  theorem  of 
algebra  that  every  periodic  single-valued  monogenic  function  of  z  is 
transcendental. 

Suppose  w  satisfies  an  irreducible  algebraic  equation  F(z,  w")  =  o,  that  is, 
an  equation  which  cannot  be  decomposed  into  the  product  of  several  factors 
of  a  similar  kind  but  of  lower  degree  in  the  variables.  Let  this  equation  be 
of  the  mih  degree  in  z  and  of  the  ;/th  degree  in  w  and  let  us  consider  the 
equation  f(z,  w0)  =  o.  This  equation  cannot  hold  for  every  value  of  z  since 
the  function  F(z,  w)  is  not  divisible  by  w  —  WQ.  Thus  at  most  can  m  values 
of  2  belong  to  the  value  WQ  of  y"(z).  But  this  contradicts  the  condition  that 
the  equation  w0  =f(z}  has  innumerable  roots,  namely,  all  of  the  form  z  —  ZQ 
-j-  kp  where  k  is  any  arbitrary  integer.  Thus  f(z}  cannot  be  an  algebraic 
function  of  z. 


MISCELLANEOUS   EXAMPLES  283 

32.    Interpret  geometrically  the  following  limit  : 


assuming  for  the  function  w  =  u  4-  iv  =f(z)  that  //  and  v  are 
continuous  and  have  continuous  first  derivatives  and  that  /x  is 
real  and  positive. 

HINT.  —  Take  two  points  zi  =z0  +  //  and  z0  on  a  curve  in  the  s-plane  and 
two  points  w\  and  w0  in  the  w-plane  corresponding  to  them  by  the  function 
w  =/(=).  Show  that  a  geometrical  interpretation  of  the  above  limit  is,  that 
the  angle  between  any  two  curves  in  the  s-plane  has  the  ratio  i  :  /j.  to  its  cor 
responding  angle  in  the  w-plane. 


33.    Compute    C-y<**  +  **y  and    C^ 

Jc          3*+f  Jc 


/ 

in  the  parameter  form  for  the  ellipse  C  about  the  origin,  x  =  a 
cos  /,  y  =  b  sin  /. 

34.    Show  by  the  CAUCHY  process  that 

o  if  ;/  is  odd, 

^ .  c  •••//—  i 

0    J  -  2  TT  if  n  is  even. 


2.  4.  6  •  •  •  n 
.    i 

HINT.  —  Put  cos  /  =  **  +  e      = where  w  =  eft  and  evaluate  the  in- 

2  2 

tegral  along  the  unit  circle  in  the  w-plane. 

35.    Show  by  evaluation  along  suitable  contours  that 


£ 
£ 

I 


I   +  X1 


dx  =  TT  e~m, 


I   +  XT          2 

sin  x 


=  -  e  m,  and 


, 
dx  =  -  . 


x  2 

Cf.  GOURSAT,  Coitrs  d' analyse,  Vol.  II,  p.  112. 


CHAPTER   V 

MAJSTY-VALUED  ANALYTIC  FUNCTIONS  OF  A  COMPLEX 
VARIABLE 

§  64.    Preliminary  Investigation  of  the  Change  of  Amplitude  of 
a  Continuously  Changing  Complex  Quantity 

Before  studying  many-valued  functions  of  a  complex  variable, 
some  attention  must  be  given,  as  suggested  in  §  4,  to  an  expres 
sion  which  has  several  values  corresponding  to  one  value  of  the 
argument  but  which  is  not  a  regular  function  of  this  argument. 
We  recall  from  §  4  that  every  complex  number 

(1)  z  =  x  -\-  iy  =  r(cos  <£  +  /'  sin  <£) 

has  infinitely  many  values  of  the  amplitude  <£,  all  of  which  are 
obtained  from  any  one  of  them  by  the  addition  or  subtraction  of 
arbitrary,  integral  multiples  of  2  TT.  From  these  infinitely  many 
values,  the  principal  value  of  the  amplitude  is  now  defined  as 
follows  : 

I.  The  principal  value*  of  the  amplitude  of  a  complex  number 
is  that  one  of  its  values  which  satisfies  the  conditions 

(2)  -7T 


It  is  essential  here  to  make  clear  that  this  principal  value  of 
the  amplitude  is  in  general,  but  not  without  exception,  a  con 
tinuous  function  of  the  real  variables  x  and  y.  Thus  let  (x^  y\} 
be  a  point,  fa  the  principal  value  of  its  amplitude,  and  let 

*  The  principal  value  of  the  amplitude  is  indicated  by  a  capital,  as  An?0. 
-  S.  E.  R. 

284 


§  54-   AMPLITUDE  OF  A  CHANGING  COMPLEX  QUANTITY      285 
fa  4-  £,  }\  +  r;)  be  a  neighboring  point.     If  we  now  put 
(   )  X,  4- 


XL  +  O'i  n 

and  if  0  is  understood  to  be  the  principal  value  of  the  ampli 
tude  of  the  expression  on  the  left,  then  as  £  and  77  vanish  0  also 
vanishes.  One  value  of  the  amplitude  of  xl  4-  £  4-  i(y\  4-  17)  is 
then  02  =  0!  4-  0-  If  0i  is  not  =  *">  then  0  can  be  taken  so 
small  that  02  also  satisfies  the  inequality  (2);  02  is  therefore  a 
principal  value,  and  the  difference  of  the  principal  values  02  and 
0!  is  only  indefinitely  small  ;  in  other  words  : 

II.  The  principal  value  of  tJie  amplitude  of  a  complex  number 
is  a  continuous  function  of  its  components  in  every  domain  of  the 
plane  which  is   not  intersected  by  the  half-axis  of  negative  real 
numbers. 

But  if  0!  =  TT,  then  $i  +  0  satisfies  the  inequality  (2)  for 
indefinitely  small  negative  values  of  0,  and  is  therefore  a  princi 
pal  value  ;  but  for  9  positive  and  indefinitely  small,  0!  +  6  is 
not  a  principal  value,  but  0!  +  $—  2ir=  —  TT  +  0  is  such  a 
value.  Theorem  II  is  therefore  extended  by  the  addition  of  the 
following  corollary  : 

III.  The  continuity  of  the  principal  value  of  the  amplitude  is 
interrupted  along  the  half  -ax  is  of  negative  real  numbers  in  so  far  as 
its  value  at  a  point  of  this  half-axis  coincides  with  the  limit  of  its 
values  at  points  adjacent  to  it  in  the  "upper"  half  -plane,  but  is 
greater  by  2  TT  than  the  limit  of  the  values  which  it  has  at  points 
adjacent  to  it  in  the  "  lower"  half  -plane. 

However,  these  latter  values  follow  continuously  from  those 
values  of  the  amplitude  of  the  negative  real  numbers  which 
=  —  TT,  and  consequently  are  less  by  2  TT  than  the  principal 
value  -f-  TT. 


286  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

What  was  said  about  the  principal  value  in  Theorems  II  and 
III  is  at  once  applicable  to  the  other  values  of  the  amplitude. 
Naming  that  value  which  is  greater  than  the  principal  value  by 
2  k-n,  the  value  of  order  k  (the  principal  value  being  thus  of 
order  zero),  we  have  the  following  theorem : 

IV.  The  value  of  order  k  of  the  amplitude  of  a  complex  number 
is  a  continuous  function  of  its  components  outside  of  the  half -axis  of 
negative  real  numbers ;  but  its  values  along  this  axis  in  the  lower 
half-plane  follow  continuously  the  values  of  order  (k  —  i)  at  the 
same  place. 

Further : 

V.  A  continuous  transition  from  the  value  of  order  k  to  that  of 
any  other,  say  to  the  value  of  order  /,  is  possible  at  no  other  place 
than  along  this  half -axis. 

For,  when  the  value  of  order  k  at  z2  differs  infinitesimally 
from  the  value  of  order  k  at  a  point  ^  indefinitely  near,  it  cannot 
at  the  same  time  differ  infinitesimally  from  the  value  of  order  /  at 
zl5  which  is  different  from  it  by  the  finite  quantity  (/  —  K]  2  TT. 

(All  values  of  the  amplitude  are  completely  undetermined  at 
z  =  o ;  the  point  z  =  o  does  not  belong  to  the  domain  for  which 
the  amplitude  function  is  defined.) 

The  conclusion  from  all  of  this  is,  —  and  it  is  the  most  im 
portant  result  of  this  investigation  : 

VI.  To  make  the  amplitude  a  continuous  function  of  position  in 
the  plane,  we  give  up  the  notion  that  it  is  single-valued  and  combine 
its  totality  of  values  into  an  infinitely  many-valued  function. 

If  two  points  zQ  ZL  of  the  plane  are  connected  by  a  given  curve, 
we  state  the  following  problem  : 

Some  one  of  the  values  of  the  amplitude  belonging  to  ZQ  is  selected; 
we  wish  to  determine  that  value  of  the  amplitude  belonging  to  zl 


§  54-   AMPLITUDE  OF  A  CHANGING  COMPLEX  QUANTITY      287 

when  a  variable  z  is  allowed  to  take  on  continuously  all  the  values 
on  the  given  curve  and  its  amplitude,  starting  with  the  given  initial 
value,  changes  continuously  as  a  result  of  this. 

The  previous  results  give  a  solution  of  this  problem,  which  is 
most  simply  exhibited  if  we  assign  a  definite  direction  from 
o  to  —  oo,  to  the  half-axis  of  negative  real  numbers,  so  that  the 
upper  half-plane  (in  which  the  coefficient  of  /  is  positive)  lies  to 
the  right,  the  lower  half-plane  to  the  left,  of  this  axis.  Therefore  : 

VII.  Provided  the  assigned  path  does  not  cross  the  half -axis  of 
negative  real  numbers,  the  value  of  the  amplitude  always  has  tJie 
same  order :  but  whenever  the  curve  crosses  this  axis  once,  tJie  value 
of  the  amplitude,  passes  to  the  next  higJier  or  to  the  ntxt  lower  order 
according  as  the  path  crosses  from  right  to  left  or  from  left  to  right. 

The  special  case  of  this  theorem  in  which  zl  coincides  with  ZQ 
merits  particular  attention  ;  it  is  stated  in  the  following  form : 

VIII.  If  z  changes  its  amplitude  continuously  in  describing  a 
closed  path,  then  the  amplitude  is  finally  greater  by  (p  —  q)  2tr  than 
before,  provided  tJie  path  crossed  the  half -axis  of  negative  real  num 
bers  p  times  from  right  to  left  and  q  times  from  left  to  right. 

But  this  formulation  is  not  yet  general,  inasmuch  as  it  em 
bodies  the  consideration  of  the  half-axis  of  negative  real  num 
bers  which  in  itself  has  nothing  at  all  to  do  with  the  problem 
and  which  has  been  introduced  only  by  our  arbitrary  definition 
of  principal  value.  However,  this  limitation  is  removed  by 
the  following  geometrical  considerations.  Let  two  non-inter 
secting  lines  Z1?  Z2  be  drawn  from  zero  to  infinity  ;  together 
they  completely  delimit  a  region  which,  as  shown  in  the  figure, 
lies  to  the  left  of  L^  and  to  the  right  of  Z2.  Let  a  closed  path  F, 
definitely  described,  cross  Lv  in  pl  points  A  from  right  to  left, 
in  qi  points  B  from  left  to  right ;  and  Z2  in  /2  points  D  from 


288  V.    MANY-VALUED   ANALYTIC  FUNCTIONS 

right  to  left,  and  in  q^  points  C  from  left  to  right.    At  the  points 
A  and  C,  the  curve  goes  into  this  bounded  domain,  at  the  points 


FIG.  25 

B  and  D  it  goes  out  of  it.     But  it  must  go  out  of  the  domain  as 
often  as  it  has  gone  into  it ;  hence, 

/.N  f       A  +  to=A  + 

(4) 


which  leads  to  the  theorem : 

IX.  The  number  (p  —  g)  appearing  in  Theorem  VIII  has  the 
same  value  for  all  lines  running  from  the  origin  to  infinity. 

(The  limitation  made  in  the  proof  of  this  theorem,  that  Lv 
and  Z2  shall  not  intersect,  can  also  be  removed.  For,  the 
theorem  can  be  proved  as  above  for  two  curves  Z:  and  Z2,  which 
first  coincide  for  a  distance  from  the  origin  and  then  sepa 
rate.  For  two  such  intersecting  curves  Zx  and  Z2,  a  third 
one  can  then  be  assigned  which  has  with  each  of  them  at 
least  one  point  of  intersection  less  than  Zt  and  Z2  have  with 
each  other.) 

X.  We  call  this  number  the  number  of  circuits  of  the  path  F 
about  the  origin. 


§  55-    THE   RIEMANN'S   SURFACE  OF  THE  AMPLITUDE      289 


Theorem  VIII  is  then  formulated  as  follows  : 

XI.  If  z  changes  its  amplitude  continuously  in  describing  a  closed 
path,  then  the  amplitude  is  finally  2  ire  greater  than  before  where  C 
is  the  number  of  circuits  of  the  path  about  the  origin. 

From  this  special  case  treated  in  the  Theorems  VIII-XI,  it 
is  now  easy  to  return  to  the  general  case  of  Theorem  VII ;  for, 
we  can  replace  any  arbitrary  path 
zbozi?  connecting  a  point  ZQ  to 
another  z{  by : 

1.  A  definite  path  z0/?2i,  for  ex 
ample,  such  a  one  which  does  not 
cut  the   half-axis  of  negative  real 
numbers ; 

2.  The     closed     path     z^zQaz^ 
which  is  composed  of  this  definite 
path  (i)  running  in  the  opposite 
direction  and  the  given  path  Z^OLZ^ 
These  remarks  are  not  limited  to 

the  investigation  of  the  amplitude  but  are  true  in  general ;  they 
are  formulated  as  follows  : 

XII.  The  change  in  value  which  a  many-valued  function  of  a 
point  undergoes  while  this  point  changes  continuously  in  tracing  an 
ARBITRARY  PATH  FROM  Z0  TO  Z1?  can  be  determined  whenever  the 
change  in  value  of  the  function  for  a  DEFINITE  PA  TH  FROM  ZQ  TO  Zx 
and  for  an  ARBITRARY  CLOSED  path  is  known. 

§  55.   The  RiEMANN'S  Surface  of  the  Amplitude 
A  clear  geometrical  representation  of  the  relations  treated  in 
the  previous  paragraph  is  obtained  by  using  the  values  of  the 
amplitude  at  ever)7  point  of  the  (x  -\-  ty)-plane  as  ordinates  per 
pendicular   to  this  plane ;    the    end-points   of   these    ordinates 


FIG.  26 


V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

determine  a  definite  surface.  We  call  the  third  coordinate  £,  in 
a  system  of  space  coordinates  of  which  two  axes  coincide  with 
our  x-  and  jy-axes  ;  then  the  coordinates  of  the  points  of  this  sur 
face  are  expressed  by  two  parameters  as  follows : 

(i)  x  =  rcos(f),   y=rsir\(f>,    £  =  </>. 

These  are  the  equations  of  a  surface  well  known  in  analytic 
geometry ;  it  is  called  the  ordinary  straight  line  helicoid ;  *  but 
these  equations  are  understood  at  present  in  a  sense  somewhat 
different  from  that  in  analytic  geometry.  There  r  and  <£  are 
regarded  as  unlimited,  real  variables  ;  all  of  the  straight  lines 
whose  equations  are  obtained  from  equations  (i)  by  giving  a 
certain  value  to  <£  and  allowing  r  only  to  vary  lie  entirely  on  this 
helicoid.  But  in  the  present  case  r  is  essentially  positive ;  our 
surface,  therefore,  contains  only  one  of  the  two  rays  into  which 
each  of  these  straight  lines  is  divided  by  their  point  of  inter 
section  with  the  £-axis.  However,  we  shall  retain  the  name 
"  helicoid  "  for  the  surface  in  the  present  case. 

The  amplitude  <£  is  thus  a  single-valued  function  of  position  on 
this  surface  since  there  is  one  and  only  one  value  of  <j>  for  each 
point  of  the  surface.  Moreover,  there  is  a  continuous  change 
of  amplitude  corresponding  to  a  continuous  progression  upon 
the  surface.  To  determine  what  final  value  of  the  amplitude 
is  obtained  at  %,  when  we  follow  a  definite  curve  starting 
from  ZQ  with  a  certain  initial  value  <£0  and  when  the  amplitude 
thus  changes  continuously,  it  is  only  necessary  to  erect  a  cylin 
der  f  on  this  curve  and  extend  it  to  intersect  the  surface.  If 
the  curve  of  the  z-plane  does  not  go  through  the  origin,  and  if 
it  has  no  double  point,  then  the  curve  of  intersection  of  the 
cylinder  with  this  surface  is  divided  into  separate  branches 

*  Sometimes  called  screw  surface.  —  S.  E.  R. 

f  Whose  element  is  parallel  to  the  £-axis  —  here  a  right  cylinder.  —  S.  E.  R. 


§  55-    THE   RIEMANN'S   SURFACE  OF  THE  AMPLITUDE      29 1 

which  have  no  point  in  common  and  are  everywhere  separated 
from  each  other  by  vertical  distances  equal  to  2  IT.  If  we  then 
follow  on  the  surface  the  branch  of  the  curve  starting  from 
C*o>  Jo»  £o  =  <&>)>  we  shall  never  trespass  on  another  branch  of  the 
curve  if  we  always  proceed  (not  by  bounds  but)  continuously 
along  the  curve.  We  arrive  finally  at  a  definite  point  of  the 
surface  lying  over  zv ;  its  ordinate  represents  then  the  desired 
final  value  of  the  amplitude.  If  the  given  curve  of  the  s-plane 
intersects  itself,  then  the  parts  of  the  curve  made  by  the  inter 
section  of  the  cylinder  with  the  surface  intersect ;  moreover, 
the  correspondence  of  the  branches  is  at  once  evident  if  we 
notice  how  the  separate  branches  of  the  curve  starting  from  the 
point  of  intersection  on  the  surface  correspond  to  the  separate 
branches  in  the  plane. 

This  method  of  representation  is  now  developed  further. 
Complex  variables  were  first  interpreted  in  the  plane ;  later,  in 
§  13,  chapter  two,  the  sphere  was  used  ;  in  the  same  way  the  sur 
face  known  as  the  helicoid  may  be  used.  For  this  purpose  we 
merely  attach  to  each  point  of  the  helicoid  the  same  complex  value 
which  belongs  to  its  perpendicular  projection  on  the  .rj'-plane ; 
and  therefore  to  each  complex  value  z  there  belongs  not  one 
definite  point  of  the  surface  as  in  the  earlier  representations,  but 
an  infinite  number  of  points  (lying  in  a  straight  line  perpendicular 
to  the  .vjr-plane).  Every  function  of  x  and  j,  whether  it  is  single- 
or  many-valued,  is  now  considered  as  a  function  of  position  on 
the  helicoid  in  that  the  values  of  the  function  belonging  to  a 
certain  z  are  assigned  to  the  points  of  the  surface  belonging  to 
the  same  z.  These  results  are  then  expressed  as  follows : 

I.  If  the  amplitude  of  z  is  considered  as  a  function  of  position  on 
the  helicoid,  this  function  becomes  single-valued  and  continuous  by 
assigning  to  each  point  of  tJie  surface  that  value  of  the  amplitude 
which  is  equal  to  its  ordinate. 


292  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

And  finally:  in  formula  (i)  the  pitch  of  the  helicoid  is  taken 
equal  to  2  TT.  But  the  size  of  the  pitch  is  evidently  arbitrary ; 
it  can  be  decreased  by  decreasing  the  ordinates  of  all  the  points 
of  the  surface  in  the  same  ratio.  It  can  finally  be  made  indefi 
nitely  small ;  the  entire  surface  is  then  composed  of  an  infinite 
number  of  flat,  thin  sheets  placed  one  upon  the  other  indefi 
nitely  close  and  connected  at  the  origin  in  the  same  manner  as 
the  sheets  of  the  helicoid  first  considered. 

II.  Such  a  surface,  composed  of  a  number  of  smooth,  flat  sheets 
connected  in  a  definite  manner,  is  called  a  plane  R  IE  MANN'S  surface. 
The  one  considered  here  has  an  infinite  number  of  sheets  extended 
over  the  whole  z-plane.     Its  sheets  are  all  connected  with  each 
other  at  the  point  z  =  o ;  this  point  z  =  o  is  therefore  for  this 
surface    a   branch-point*    of  infinitely  high   order.     Over   every 
other  point  of  the  s-plane  (even  over  the  points  of  the  half-axis 
of  negative  real  numbers)  the  sheets  remain  separate  and  are 
arranged  simply  one  upon  another. 

The  same  surface  is  also  obtained  in  another  way  as  follows : 
we  cut  the  z-plane  along  the  half-axis  of  negative  real  numbers 
from  o  to  —  oo.  Let  us  consider  an  infinite  number  of  such 
s-planes  cut  in  this  way,  and  let  us  number  them  by  an  index  k 
which  takes  all  integral  values  from  —  oo  to  +  oo.  Let  us  now 
arrange  them  one  upon  another,  so  that  the  (k-\-  i)th  sheet  is 
the  next  above  the  /£th.  Finally  let  us  connect  the  right  bank 
of  the  cut  in  the  k\h  sheet  with  the  left  bank  of  the  cut  in  the 
(k+  i)th  sheet. 

III.  Upon   this   surface,    constructed    in    either    manner,    the 
values  of  the  amplitude  are  thus  arranged  as  a  single-valued  and 

*  That  is,  if  a  point  z  makes  a  complete  circuit  of  a  point  P  in  the  z-plane  and 
returns  to  its  original  position,  and  in  so  doing  the  value  of  w  (the  function)  is 
always  changed,  then  the  point  P  is  called  a  branch-point.  As  an  illustration  of 
how  the  function-values  pass  into  one  another  on  describing  closed  paths  around 
a  branch-point,  cf.  Exs.  i,  3,  5,  end  of  §  59.  —  S.  E.  R. 


§  55-    THE  RIEMANN'S  SURFACE  OF  THE  AMPLITUDE      293 

continuous  function  of  position;  and  this  is  the  final  result  of  the 
discussion. 

We  shall  frequently  have  occasion  in  what  follows  to  use 
such  "  RiEMANN'S  surfaces  "  to  represent  graphically  the  rela 
tion  between  the  different  values  of  a  many-valued  function. 
In  this  connection  it  seems  most  practical  to  think  of  the  sur 
face  as  extended  over  the  sphere  and  not  over  the  plane ;  such 
a  representation  is  obtained  by  projecting  the  plane,  that  is,  the 
surface  spread  out  upon  it,  stereograph ically  (§  13)  upon  the 
sphere.  This  is  of  no  particular  use  in  the  case  just  consid 
ered  ;  however,  in  this  transformation  from  the  plane  to  the 
sphere  we  observe  that  the  half-axis  of  negative  real  numbers 
corresponds  to  a  half-meridian  which  connects  the  points  O  and 
Of.  We  notice  too  that  the  sheets  are  connected  at  the  latter 
point  just  as  at  the  former,  with  this  difference  however,  that 
if  we  regard  the  sheets  about  O  as  "wound  right-handed,"* 
then  those  about  O'  are  "  wound  left-handed.1'  For,  a  line  upon 
the  sphere  which  encircles  the  point  z  =  o  in  the  positive  sense, 
that  is,  so  that  this  point  always  lies  to  the  left  in  passing  along 
the  curve,  has  at  the  same  time  the  point  at  infinity  to  the  right 
and  encircles  it  therefore  in  the  negative  sense. 

A  further  explanation  is  necessary  in  order  to  avoid  misun 
derstandings  that  might  otherwise  arise.  In  the  above  para 
graphs  we  have  frequently  spoken  of  "  sheets  "  of  the  surface  ; 
this  was  due  chiefly  to  the  way  in  which  the  surface  was  con 
structed  from  planes  cut  along  the  half-axis  of  negative  real 
numbers ;  and  therefore  upon  the  arbitrary,  fixed  definition  of 
the  principal  value  of  the  amplitude.  The  joining  of  the  sheets 
at  the  cut  is  not  visible  on  the  completed  surface ;  but  such 
connection  and  the  resulting  individual  sheets  become  evident 
by  supposing  an  arbitrary,  vertical  cut  through  all  the  sheets 
*  As  is  customary  in  technics  but  different  in  botany. 


2Q4  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

running  from  o  to  oo.  We  notice  too  that  two  points  which 
are  vertical  to  the  same  point  in  the  plane  and  which  lie  in  dif 
ferent  sheets  for  one  such  cut  lie  in  different  sheets  for  any  such 
cut.  But,  given  two  points  of  the  surface  which  are  situated 
above  different  points  of  the  plane,  we  can  then  choose  the 
position  of  the  cut  so  that  these  points  lie  in  the  same  sheet  or 
in  different  sheets  of  the  surface.  Hence  the  expression  "  two 
points  of  the  same  sheet'1'1  always  has  a  definite  meaning  only  with 
regard  to  a  definite  cut  previously  chosen  (cf.  end  §  59). 

§  56.   The  Logarithm 

The  value  of  the  integral 

JTf< 

according  to  VI,  §  35,  is  a  regular  function  of  its  upper  limit 
inside  of  every  simply  connected  domain  which  contains  within 
it  the  point  z  =  i  but  neither  the  origin  nor  the  point  at  infinity, 
—  provided  that  the  path  of  integration  also  lies  entirely  in  the 
domain.  (The  origin  and  the  point  at  infinity  must  here  be 
excluded,  since  the  function  to  be  integrated  has  a  pole  in  the 
first  case,  and  while  the  function  remains  regular  in  the  second 
case  it  is  not  zero  of  order  higher  than  the  first;  cf.  IV,  §  45.) 
If  z  is  real  and  positive,  and  if  the  axis  of  positive  real  num 
bers  is  chosen  as  the  path  of  integration,  then  the  value  of  in 
tegral  (i)  is,  as  is  well  known,  equal  to  the  natural  logarithm 
of  z.  We  retain  here  this  name  and  the  corresponding  symbol 
for  the  function  for  the  case  where  z  is  a  complex  number ;  we 
define  accordingly : 

I.  The  natural  logarithm  of  a  complex  number  z,  log  z,  is  any 
one  of  the  values  which  integral  (/)  takes  on  when  the  path  of  inte 
gration  is  arbitrary. 


§  56.    THE   LOGARITHM 


295 


The  determination  of  the  values  of  the  logarithm  of  a  com 
plex  number  is  made  to  depend  upon  functions  of  real  variables 
known  in  elementary  analysis,  by  representing  the  complex  num 
bers  in  terms  of  their  absolute  values  and  amplitudes  as  in  §  4. 
For  this  purpose  we  put 

(2)  z  =  ;-(cos  <p  +  *  sin  <£) 

(3)  £  =  p(cosi^  +  /sin^r). 

To  discuss  the  simplest 
case  let  us  take  as  the 
required  path  of  inte 
gration  a  piece  of  the 
axis  of  real  numbers 
from  i  to  \z\  and  an 
arc  of  a  circle  whose 
center  is  at  the  origin 
and  which  connects  the 


and  z  (Fig. 


FK;.  27  a 


points 

27).    Along  the  first  part  of  this  path  1^  =  0,  £  =  p,  d£  =  dp  and 
p  takes  on  all  the  values  from  i  to  \  z  \ .    For  this  part  of  the  path 

the  following  integral, 

(4) 


0 


1*1 

FIG. 


taken  along  the  real 
path  between  the  real 
limits  is,  therefore,  a 
special  case  of  the  inte 
gral  (i);  and  Log  |  z  \  is 
here  understood  to  be 


*  The  capital  here  indicates,  as  in  Ex.  3  at  the  end  of  §  56,  a  definite  "  branch  " 
of  the  logarithm  called  the  principal  value  of  the  logarithm  (cf.  IV).  In  what  fol 
lows  it  will  be  so  written.  —  S.  E.  R. 


2 96  V.    MANY- VALUED   ANALYTIC   FUNCTIONS 

the  real  natural  logarithm  of  the  real  positive  number  z  \ ,  defined 
in  elementary  mathematics.  If  p  is  constant  and  =  |  z  \  along 
the  second  part  of  the  path  of  integration,  then, 

(5)  J£  =  |  z  \  (  -  sin  $  +  /  cos  $)<ty  =  iftty, 

and  i/'  takes  on  all  the  real  values  from  o  to  <£.  For  the  second 
part  of  the  path  we  therefore  have  a  special  case  of  the  integral 
(i)  equal  to  i  times  the  integral 


(6)  f  %  = 

Jo 


taken  along  the  real  path.  This  integral  defines  <f>  ;  for  <£  we 
therefore  take  that  value  of  the  amplitude  of  z  which,  according 
to  §  54,  is  obtained  when  z,  starting  from  |  z  \  ,  traces  the  pre 
scribed  arc  of  a  circle,  and  when  the  amplitude  thus  starting 
from  o  changes  continuously.  Every  value  of  the  amplitude 
can  therefore  be  obtained  by  allowing  the  prescribed  arc  of  a 
circle  to  include  more  than  a  whole  circumference. 

The  result  thus  found  for  this  special  kind  of  path  of  integra 
tion  is  true  generally.  For,  every  arbitrary  preassigned  path 
from  i  to  z  may  be  deformed,  without  passing  through  the  origin 
or  through  infinity,  to  a  path  of  the  kind  just  considered.  It 
therefore  follows,  according  to  V,  §  35,  that  the  values  thus 
found  represent  the  totality  of  the  values  of  log  z  determined  by 
definition  (i).  The  results  of  the  investigation  are  expressed 
completely  as  follows  : 

II.  The  totality  of  values  of  the  logarithm  of  the  complex  number 
z  =  r  •  e^  is  given  by  the  formula 

(7)  logs 


in  which  Log  r  is  the  real  logarithm  of  the  absolute  value  of  z,  and 
<f>  is  an  arbitrary  value  of  its  amplitude, 


§  56.    THE   LOGARITHM  2Q/ 

III.  The  logarithm  of  a  complex  variable,  as  it  is  defined  by  (/), 
is  thus  an  infinitely  many-valued  function,  the  totality  of  whose 
values  is  obtained  from  any  one  of  them  by  the  addition  of  arbitrary 
integral  multiples  of  2  iri. 

IV.  By  using  the  principal  value  of  the  amplitude  a  definite 
"branch  "  of  this  infinitely  many-valued  futution  is  obtained ;  it  is 
called  tlie  principal  value  of  the  logarithm* 

A  real  positive  number  is  a  particular  case  of  a  complex 
variable.  As  such  it  therefore  has  infinitely  many  logarithms 
in  the  sense  defined  here  ;  of  these  the  principal  value  is  identi 
cal  with  the  real  logarithm  defined  in  an  elementary  way,  the 
others  have  imaginary  parts  which  are  even  multiples  of  vi. 
The  imaginary  parts  of  logarithms  of  negative  real  numbers  are 
odd  multiples  of  iri. 

V.  As  the  basis  for  representing  the  logarithm    as   a    single- 
valued  function  of  position  we  use,  therefore,  the  RIEM ANN'S  sur 
face  studied  in  the  previous  paragraph. 

It  is  essential  that  we  study  now  the  most  important  proper 
ties  of  the  logarithmic  function  as  defined.  The  first  of  these 
properties  is  that  each  of  its  branches  is  regular,  according  to 
VI,  §  35,  in  every  domain  which  lies  entirely  in  the  finite  part 
of  the  plane,  which  is  simply  connected,  and  which  does  not 
contain  the  origin  within  it ;  it  can  then  be  developed  in  a  TAY 
LOR'S  series  in  the  neighborhood  of  every  point  excepting  only 
o  and  oo  .  The  coefficients  of  this  development  are  determined 
from  the  defining  equation  (i)  by  successive  differentiation  ;  it 
thus  follows  that 
(8) 


*  Thus,  that  value  of  log  [z  =  r(coscj>  +  *'sin  </>)]  for  which  —  n<4><  it  is  writ 
ten  Log  z  =  Log(r  =  |  z  \  )  +  /  Am  s.—  S.  E.  R. 


298  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

just  as  when  z  is  real.     We  observe  in  particular  : 

VI.    The   development  of   the  principal  value   in   powers   of 
(z-  i)  is 


The  elementary  logarithm  of  a  positive  real  number  has  also 
the  fundamental  property  that 

(IO)  log  (S^)  =  log  Zi  +  log  ZZ. 

In  order  to  investigate  whether  and  in  what  sense  this  property 
also  holds  for  the  infinitely  many-valued  function  of  complex 
argument  designated  here  by  the  name  logarithm,  we  start  from 
the  fact  that  each  preassigned  path  from  i  to  Z&  can  be  so  de 
formed  as  to  make  it  pass  through  the  point  %  without  in  this 
way  changing  the  value  of  the  integral  : 


—  nj&  *1*2- 


This  integral,  for  all  of  its  values,  can  now  be  written  in  the 
form  of  the  sum  : 


rf+.t' 


by  suitably  choosing  the  two  paths  of  integration.  The  first 
one  of  these  integrals  is  a  value  of  log  %  ;  let  us  introduce  a  new 
variable  of  integration  77  in  the  second  by  the  substitution  : 

(13)  f  =  M- 

We  have  investigated  this  substitution  in  §  9  ;  it  is  reversibly 
unique  over  the  whole  plane.  To  the  path  from  ^  to  %  £2  pre 
viously  determined  in  the  £-plane,  there  corresponds  then  point 
for  point  in  the  ^-plane  a  definite  path  from  77  =  i  to  77  =  zz  ; 


§  56.    THE  LOGARITHM  299 

and  therefore  the  second  integral  of  (12)  may  be  replaced  by 

a  value  of  log  z2. 


p*. 

Jl         Tfl 


i 

Thus  the  validity  of  (10)  for  complex  arguments  is  proven  in 
the  sense  that  if  any  value  is  assigned  to  the  left-hand  side  we 
can  always  so  choose  the  values  of  the  logarithms  on  the  right- 
hand  side  that  the  equation  is  satisfied.  We  can  even  choose 
arbitrarily  the  value  of  one  of  the  two  logarithms  on  the  right- 
hand  side  and  then  determine  the  other  so  that  the  equation 
remains  true.  For,  when  a  path  from  i  to  S&  and  one  from 
i  to  zl  are  agreed  upon,  another  path  from  zv  to  z&  can  always 
be  so  determined  that  all  three  paths  together  form  a  closed 
curve  which  encircles  the  origin  zero  times  (X,  §  54). 

Conversely  each  value  of  the  right-hand  side  of  (10)  is  equal 
to  a  value  of  the  left-hand  side.  For,  suppose  arbitrary  paths 
from  i  to  zv  and  from  i  to  z^  are  given  ;  by  means  of  the  substi 
tution  (13)  a  definite  path  from  %  to  Z&  corresponds  to  the 
path  from  i  to  &,,  and  this  then  combined  with  the  path  from  i 
to  %  gives  a  definite  path  from  i  to  %s2.  On  this  account 
therefore, 

VII.  Equation  (10)  zV  true  for  complex  arguments  z  in  the  sense 
that  every  value  of  the  right-hand  side  is  equal  to  a  value  of  the  left- 
hand  side  and  conversely,  and  that  then  the  totality  of  values  of  the 
two  sides  coincide.* 

Having  once  determined  the  equality  of  any  values  whatever 
of  the  two  sides,  we  might  have  derived  therefrom  that  both 
sides  of  the  equation  have  the  same  degree  of  many-valuedness, 
since  for  both  sides  the  transition  from  one  value  to  any  other 
takes  plfiCe  by  the  addition  of  arbitrary  integral  multiples  of 

*  That  is,  it  is  a  complete  equation,  or  one  which  is  completely  true. —  S.E.  R. 


300    v.  MANY- VALUED  ANALYTIC  FUNCTIONS 

2  TTZ.  The  method  pursued  shows  further  how  the  third  path  is 
to  be  chosen,  when  two  paths  of  integration  are  given,  in  order 
to  satisfy  the  equation. 

In  other  equations  between  many-valued  functions  the  con 
ditions  may  be  entirely  different.  If,  for  example,  we  put 
zl  =  z2  =  z  in  equation  (10),  we  conclude  that  the  two  paths  on 
the  right-hand  side  should  coincide  (or  at  least  encircle  the 
origin  equally  often),  and  it  then  follows  from  the  first  proof  of 
Theorem  VII  that  in  the  resulting  equation 

(14)  logO2)  =  2  logs 

every  value  of  the  right-hand  side  is  equal  to  a  value  of  the  left- 
hand  side.  But  if  we  prescribe  the  path  from  i  to  zz  and  one  of 
the  paths  from  i  to  z  we  can  not  conclude  that  the  path  from  z  to 
z2,  compounded  from  the  return  path  from  z  to  i  and  the  path 
from  i  to  22,  is  transformed  by  the  inverse  of  substitution  (13)  into 
a  second  path  from  i  to  z  which  coincides  with  the  prescribed 
one  from  i  to  z  or  which  may  be  reduced  to  it  without  going 
through  the  origin.  From  the  second  method  we  see  that  the 
left-hand  side  of  (14)  is  determined  only  for  integral  multiples 
of  2  IT/,  the  right-hand  side  only  for  such  multiples  of  4  tri.  We 
find  accordingly  that : 

VIII.  In  equation  (14)  every  value  of  the  right-hand  side  is  equal 
to  a  value  of  the  left-hand  side,  but  the  left-hand  side  may  have  the 
values  of  ^  log  z  +  2  -iri  in  addition  to  this. 

It  is  important  to  notice  also  that  equations  (10)  and  (14) 
are  not  always  true  if  we  use  only  the  principal  values  of  all 
the  logarithms  in  them,  as  simple  values  show  (put;  for  exam- 

3»rt 

pie,  z  =  e*  in  (14)). 


§  56.    THE   LOGARITHM  30  1 

EXAMPLES 

1.  Any  value  of  log  z  is  a  continuous  function  of  both  x  and 
_>',  except  when  x  =  o,  y  =  o.     Prove. 

2.  Show  that  in  the  equation 

logfe)  =  log  %  +  log  z2 
every  value  of  either  side  is  one  of  the  values  of  the  other  side. 

HINT.  —  Put  21  =  ri(cos  6\  +  i  sin  0i)  and  z»  =  r2(cos  02  +  *  sin  02),  and 
apply  the  formula. 

3.  Show  that  Log(sI22)  =  Log  zl  +  Log  z2,  is  not  true  in  all 
cases. 

For  example  if  zl  =  z.2  =  i/2(—  i  +  /\/3)  =  cos  ^-  +  /  sin  —  ^, 

3  3 

then  Log  S!  =  Log  s2  =  •§•  ""^  an<^  Log  Sj  +  Log  %  =  |  ?r/,  which  is 
one  of  the  values  of  log  (s^o),  but  not  the  principal  value. 

What  is  the  value  of  Log  (%&)  for  the  special  value  of  % 
and  s2?  Ans.    (—  2/^)(iri). 

4.  Show  that  in  the  equation 

log  zm  =  m  log  z,  m  being  an  integer, 

every  value  of  the  right-hand  side  is  a  value  of  the  left-hand 
side,  but  that  the  converse  is  not  true.  What  values  belong  to 
the  left-hand  side  of  this  equation  that  are  not  values  of  the  right- 
hand  side  ? 

5.  Is  the  equation  of  Ex.  3  above  true  if  the  line  from  zl  to 
z2  cuts  the  negative  half  of  the  real  axis  ? 

6.  Show  that  the  equation 

=  Log  (z  ~  ^  -  Log  (z  ~  b"> 


is  true  if  z  lies  outside  of  the  domain  bounded  by  the  line  join 
ing  the  points  z  =  a  and  z  =  b  and  lines  through  these  points 


3<D2  V.    MANY- VALUED   ANALYTIC   FUNCTIONS 

parallel  to  the  .*-axis  and  extending  to  infinity  in  the  negative 
direction. 

7.    The  equation 

)  =  Log  (i  -  a/z]  -  Log  (i  -  b/z) 


is  true  if  z  lies  outside  the  triangle  formed  by  the  three  points 
o,  a,  b.  Prove. 

8.  If  z  =  x  +  (y,  then  log  log  z  —  Log  R  +  (0  +  2  &'ir)i  where 

JP  =  (Log  ry  +  (0  +  2  /br)2 

and  0  is  the  least  positive  angle  determined  by  the  equations, 
cos  0  :  sin  6  :  i  :  :  Log  r :  0  +  2  k-n- :  V (Log  7f  +  (0  +  2  k-nf. 

Plot  roughly  the  doubly  infinite  set  of  values  of  log  log  (i  +t  \/3), 
indicating  which  of  them  are  values  of  Log  log(i  H-iVs)  and 
which  of  log  Log  (i  +  /V3). 

9.  Is   the  equation  a6  =  (#2)3  a  complete  equation  ?     Show 
by  use  of  logarithms. 

10.  Are  the  equations 

am  (  —  i  =  am  ^  —  am  zz  and  Am  [  —  ]  =  Am  %  —  Am  z2 

W  W 

complete  equations  ? 

11.  Show  that  the  exponential  function  expz  or  e?  is  a  single- 
valued  function  of  z. 

12.  When  x  is  negative,  how  does  log  x  differ  from  log  |  x    or 
from  (1/2)  log.*2? 

13.  We   know   that  lim  \   °g(T  +w)  j.  _  z  when   w  is   real. 

«±o  1          w          j 

This  result  may  be  extended  to  complex  values  of  a/.     For, 


§  56.     THE   LOGARITHM  303 

the  path  of  integration  being  the  straight  line  from  i  to  i  -f  w. 
This  line  is  represented  by  the  equations 

x  =  i  +  /p  cos  <£,   y  =  tp  sin  <£,    o  <  t  <^  i, 

p  being  the  modulus  and  <£  the  amplitude  of  w.     Thus 
,       ,  \       Cl    P  (cos  <t>  -\-  i  sin  <$>}dt 

log  ( I  -|-  ft')  =    I  -Z* , 

Jo    i  +  /p(cos  <£  +  /  sin  <£) 


and 


=  I    - 

*/o    i 


/p(cos  <£  +  /  sin  <£) 
r1    /(cos  4>  +  t  sin 

=  I-"Jo  7T 


/p(cos  </>  -|-  /  sin 

The  modulus  of  the  last  term  is  less  than 
tdt 


which  approaches  zero  with  p.  and  hence  lim  \  —  °^T     —  ^  !•  =i. 

«=M)    I  «' 

If  w  =  //  +  /V,  and  «  and  v  each  approach  zero,  then  w  ap 
proaches  the  origin  along  a  path  the  nature  of  which  depends 
upon  the  way  in  which  u  and  v  approach  zero,  or  on  the  rela 
tions  which  hold  between  them  in  the  process.  Thus  if  //  were 
always  equal  to  v,  the  path  would  be  a  straight  line  bisecting 


the  angle  between  the  axes.     Thus  tv    approaches  i 

w 
as  w  approaches  zero. 

14.    Show    that    the    formula    —  log  <£(/)  =  <£'(/)/<£(/)    holds 

dt 

generally  when  <£  is  a  complex  function  of  the  real  variable  /. 
Put  <f>  =  //  +  iv  and  log  <£  =  (1/2)  log(//2  +  #2)  -f  *'  tan-1^/^)  and 
differentiate  according  to  the  usual  formulas. 


304  V.    MANY-VALUED   ANALYTIC  FUNCTIONS 

§  57.   Conformal  Representation  Determined  by  the  Logarithm 

We  investigate  now  the  conformal  mapping  of  the  s-plane 
upon  the  w-plane  determined  by  the  function  : 

(1)  w  =  \ogz-, 

in  this  connection  we  keep  in  mind  the  principal  value  of  the 
logarithm.  In  the  theory  of  the  real  logarithm  of  a  real  positive 
number  z\  it  is  known  that  such  a  logarithm  takes  on  real 
values  continually  increasing  from  —  oo  to  +  oo  as  z  \  increases 
from  o  to  oo  .  Further,  <£  continually  increasing  passes  from 
—  TT  to  +  TT  as  z  describes  a  circle  about  the  origin  in  the  posi 
tive  sense,  starting  at  its  intersection  with  the  negative  ^-axis 
and  returning  to  that  place.  Since  a  circle  about  the  origin  and 
a  radius  vector  starting  at  the  origin  can  intersect  in  only  one 
point,  it  follows  that : 

I .  The  principal  value 

(2)  w  =  u  -f-  iv 

of  the  logarithm  takes  on  each  finite  complex  value  at  one  and  only 
one  point  of  the  plane  providing  the  imaginary  part  v  satisfies  the 
inequality  : 

(3)  _7r<Z;^+7r. 

But,  expressed  geometrically,  this  means  that: 

II.  The  z-plane  cut  along  the  half-axis  of  negative  real  numbers 
is  mapped  conformally  by  the  principal  value  of  the  logarithm  upon 
the  parallel  strip  of  the  w-plane  bounded  by  the  lines  v  =  —  TT  and 
v  —  -h  TT. 

Thus  the  parallels  to  the  #-axis  correspond  to  the  rays  of 
the  s-plane  starting  at  the  origin,  the  parallels  to  the  z/-axis 
correspond  to  the  concentric  circles  about  the  origin  in  the 
z-plane. 


§  57-   CONFORMAL  REPRESENTATION  BY  THE  LOGARITHM     305 

Going  now  from  the  principal  value  to  the  other  values  of  the 
logarithm,  we  find  that : 

III.  The  z-plane  cut  along  the  half-axis  of  negative  real  numbers 
is  mapped  by  the  kth  value  of  the  logarithm  upon  t)iat  strip  of  tfie 
w-plane  bounded  by  the  parallels  : 

V—(2k  —  I)TT,    v  =  (2  k  +  I)TT. 

The  maps  of  the  z-plane  upon  the  w-plane  determined  by  the 
different  branches  of  the  logarithmic  function  are  therefore 
contiguous  throughout  the  w-plane  and  finally  cover  the  whole 
of  it  once  without  gaps.  From  this  it  follows  that : 

IV.  There  is  always  one,  and  only  one,  value  of  z  (finite  and 
different  from  zero],  for  which  one  of  the  values  of  log  z  is  equal  to 
an  arbitrary,  preassigned finite  complex  number  w. 

Let  us,  therefore,  consider  z  as  a  function  of  w,  that  is,  the 
problem  "  to  revert  the  logarithm'''  We  find  that  this  function 
is  single-valued  over  the  whole  plane.  It  is  furthur  continuous 
over  the  whole  plane,  as  is  seen  from  the  definition  of  the 
logarithm  by  means  of  the  definite  integral ;  moreover,  the  con 
tinuity  is  not  broken  at  the  boundaries  of  the  parallel  strips,  as 
we  see  from  the  results  of  §§  54,  55  relative  to  the  continuous 
connection  between  the  different  branches  of  the  logarithm  (or 
amplitude).  Finally,  this  function  has  a  definite  first  derivative 

over  the  whole  plane  : 
/  x  dz       i   /  dw 

(4)  ^=  /lTz=z' 

and  thus  z  is  finite  and  different  from  zero  for  all  finite  values 
of  iv.  In  consequence  of  Theorem  IX,  §  38,  and  the  definition 
of  a  regular  function,  we  therefore  have  : 

V.  The  inverse  of  the  logarithm  is  a  function  regular  over  the 
whole  plane  and  is  therefore  a  transcendental  integral  function. 


306  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

Having  obtained  this  result  we  may  now  use  the  method  of 
undetermined  coefficients  to  determine  the  coefficients  of  the 
corresponding  series  by  substituting 


in  the  differential  equation  (4).  The  following  recursion 
formula  for  An  is  thus  obtained  : 

nAn  =  An_i  ; 

and  as  AQ  must  be  equal  to  i  (since  z  =  i,  w=  o  is  a  pair  of 
corresponding  values),  we  use  this  formula  to  determine  the  co 
efficients  An  successively.  We  thus  obtain  : 

(5)  s=i+  j£^,thatis: 

VI.  The  inverse  of  the  logarithm  is  the  exponential  function  of 
complex  argument  discussed  in  §  40. 

We  might  also  have  obtained  this  result  in  many  other  ways, 
for  example,  by  reverting  the  series  (9),  §  56,  in  the  sense  of  X, 
§  46,  or  by  showing  that  the  conformal  mapping  determined  by 
the  logarithm  is  exactly  the  inverse  of  the  conformal  mapping 
determined  by  the  exponential  function.  The  method  used 
here  is  important,  since  it  can  be  used  in  complicated  cases  to 
determine  whether  a  proposed  problem  of  inversion  can  be 
solved  in  terms  of  a  single-valued  function.  To  avoid  misun 
derstandings,  we  state  further  that  it  is  not  sufficient  in  the 
proof  of  Theorem  V  to  show  that  the  inverse  function  is  regular 
in  the  neighborhood  of  every  point  of  the  domain  for  which  it  is 
defined  ;  it  is  much  more  essential  that  we  obtain  a  clear  con 
ception  of  the  form  of  this  defining  domain  ;  this  is  most  easily 
done  by  investigating  the  conformal  mapping. 


§  57-  CONFORMAL  REPRESENTATION  BY  THE  LOGARITHM    307 

EXAMPLES 

1.  For  the  transformation  w  =  log  z  find  the  curves  in  the 
z-plane  which  correspond  respectively  to  the  lines  u  =  const, 
and  v  =  const. 

If  z  =  x  +  iy  =  /'(cos  6-\-i  sin  6) 

and  w  =  u  -f-  iv  =  p  (cos  <£  +  /  sin  </>), 

then  *  =  ^u  cos  v,       u  =  log  r, 

y  =  e"  sin  z1,       v  =  6  +  2  >£, 

where  £  is  any  integer.  Describe  the  motion  of  z  while  a/  de 
scribes  the  whole  of  a  line  parallel  to  the  z>-axis. 

2.  Show  that  to  a  straight  line  in  the  w-plane  corresponds, 
by  the  transformation  w  =  \ogz,  an  equiangular  spiral   in  the 
2-plane. 

2  a.  If,  in  the  stereographic  projection  defined  in  Ex.  i ,  at 
the  end  of  §  13,  we  introduce  a  new  complex  variable 

w  =  u  +  iv  —  —  i-  \og(z/2)  =  —  i-  log  [(i/2)(^ -f- /»] 

n 

so  that  u  =  <j>,  v  =  log  cot  - ,  we  obtain  another  map  of  the  sur 
face  of  the  sphere  usually  called  MERCATOR'S  Projection.  On 
this  map  parallels  of  latitude  and  longitude  are  represented  by 
straight  lines  parallel  to  the  axes  of  u  and  v  respectively. 

NOTE. — The  problem  of  making  maps  of  the  earth's  surface  by  applying 
the  principles  of  stereographic  projection  and  conformal  representation  is  of 
great  interest.  The  discovery  of  the  compass  brought  with  it  the  idea  of 
steering  a  course  making  with  all  meridians  a  constant  angle.  This  course 
was  a  spiral  and  was  called  a  rhumb  line  or  loxodroma.  If  the  earth's  sur 
face  (regarded  as  a  sphere)  be  inverted  from  any  point  of  the  surface,  say 
the  north  pole,  into  a  plane,  for  example  into  the  plane  tangent  at  the  south 
pole,  the  meridians  become  a  pencil  of  rays  through  the  origin  in  the  plane 
and  the  loxodromes  are  then,  by  isogonality,  curves  cutting  this  pencil  at  a 
constant  angle,  that  is,  equiangular  spirals.  But  the  map  so  formed  by  stereo- 


308  V.    MANY-VALUED    ANALYTIC   FUNCTIONS 

graphic  projection  was  not  sufficiently  simple  since  the  loxodromes  were  the 
important  lines.  A  map  was  wanted  on  which  the  loxodromes  would  appear 
as  straight  lines.  This  was  accomplished  by  mapping  the  inverse  of  the 
sphere  by  means  of  w  =  log  2.  And  this,  then,  is  the  principle  of  MERCATOR'S 
Projection. 

Of  the  memoirs  which  treat  of  the  construction  of  maps  of  surfaces  as  a 
special  question,  the  most  important  are  those  of  LAGRANGE,  Collected  Works, 
Vol.  IV,  pp.  635-692,  and  GAUSS,  Ges.  Werke,  Vol.  IV,  pp.  189-216.  Also  a 
treatise  by  HERZ,  Lehrbuch  der  Landkartenprojectlonen,  Teubner,  1885. 

2  b.    Discuss  the  map  determined  by  the  equation 


showing  that  the  straight  lines  for  which  x  and  y  are  constant 
correspond  to  two  orthogonal  systems  of  coaxial  circles  in  the 
ze/-plane. 

3.  Find  all  the  values  of  /•'. 

By  definition  i{  =  exp  (Y  log/). 

But  /  =  cos  -  +  /  sin  —  ,    log  /  =  [  2  kir  +  -  ]  *', 

22  V  2/ 

where  k  is  any  integer.     Thus, 

i*  -  exp  {  -  (2  k  +  1/2)71-5  =  ^(2*+1/2)7r. 
The  values  of  /*'  are,  therefore,  all  real  and  positive. 

4.  Find  all  the  values  of  (i  +  /)*',  z(1+i),  (i  +  0(1+<)- 

5.  Find  the  general  value  of  a2.     Let 

z  =  x  +  ty,    a  =  p(cosO  +  /sin0) 
where  —  TT  <  0  ^  ?r. 

By  definition  a2  =  exp  (z  log  a). 

But       z  log  a  =  (x  +  iy)  \  Log  p+(6  +  2  mir]i\  =  L  +  iAf, 
where  L  =  x  log/o  —  y(0  +  2  mir\   M=y  logp  +  x(6  +  2  mir) 
and  a*  —  exp  (z  log  a)  =  eL(cos  M+  i  sin  M). 


§  57-  CONFORM  AL  REPRESENTATION  BY  THE  LOGARITHM    309 
Therefore  the  general  value  of  a*  is 

^logp-y(0+2m,r)[-cos  |j,  I0gp  +X(0  +  2  Mlf)  \  +  I  sin  \y  log  p 


This  is  in  general  an  infinitely  many-valued  function  corre 
sponding  to  the  different  values  of  ;;/  unless  y  =  o.  But  even 
if  y  =  o  and  z  irrational,  there  are  an  infinite  number  of  values 
each  of  which  have  the  same  modulus. 

6.  Find  the  principal  value  of  a*.     (Put  ;;/  =  o  in  the  general 
formula.) 

7.  There  are  two  particular  cases  in  Ex.  5  that  are  of  interest  : 
(I)  if  a  is  real  and  positive  and  z  real,  then  p  =  a,  0  =  o,  x  =  z, 
y  =  o,  and  the  principal  value  of  a*  is  <?*loga  ;  but  (II)  if  \a\  —  i 
and  z  is  real,  then  p  =  i  ,  x  =  z,  y  =  o  and  the  principal  value 
of  (cos  0  +  z  sin  0)z   is   (cos  zQ  +  /sin  zO),  —  a  generalization   of 
DE  MOIVRE'S  theorem. 

8.  Find  the  general  value  and  also  the  principal  value  of  e*. 
(For  the  general  value  put  e  for  a  in  the  general  formula  so  that 
logp  =  i,  0  =  o.     The  principal  value  of  e*  is  e*(cos  y-{-t  s'my). 

9.  Show  that  log  (e2)  =  (i  -f  2  miri)z  +  2  tnri,  m  and  ;/  being 
any  integers,  and  that  in  general  log  (a2)  has  a  double  infinity 
of  values. 

10.    In  what  cases  are  any  of  the  values  of  x1,  where  x  is  real, 
themselves  real  t 

If  x  >  o,  then 

xx=  exp  (x  log  x)  =  Jexp  (x  Log  x)  j  (cos  2  m-n-x  -f  /  sin  2  mirx)  the 
first  factor  of  which  is  real.  The  principal  value,  for  which 

m  =  o,  is  always  real. 

p 
If,  however,  x  is  a  rational  fraction  of  the  form  -  —  -^—  ,  or  is 

irrational,  there  is  no  other  real  value.     But  if  x  is  of  the  form 


310  V.    MANY-VALUED  ANALYTIC   FUNCTIONS 

p/2  q  there  is  one  other  value,  viz.  :    exp  (x  Log  x]  given  by 
m  =  q.. 

If  x  =  —  h  (  <  o),  then 

xx  =  exp  |  —  h  log  (—  h)  \  =  [exp  (—  h  Log  >6)](cos  0  +  /sin  0) 
where  0  =  —  (2  #z  +  i)?r/i  The  only  case  in  which  any  value 
is  real  is  that  for  which  h  =  —  £•  —  ,  whence  m=>g  gives  the 

2?+I 

real  value, 

exp(-/£Log  ^){cos(—  /TT)+*  sin(—  /TT)}==(-  i)p^-fc. 

The  cases  of  reality  are  illustrated  by  the  following  examples: 


(-i/3)-*  =  -^3- 

11.  Show  that  the  real  part  of  /L°g<1+''>  is 

<p-J(«+')»t  .  cos  {1(4  £+  I)TT  log  2  J, 

where  k  is  any  integer.     How  does  this  differ  from  the  real 
part  of  1  10*1+<>  ? 

12.  The  values  #z  when  plotted  on  the  ARGAND  diagram  are 
the  vertices  of  an  equiangular  polygon   inscribed  in   an  equi 
angular   spiral    whose    angle    is    independent    of   a.      (Math. 
Trip.,  1899.) 

If  az=  rcos0  +  /sin0, 


then  r=  ^log  P-»^+^\  0  =  y  log  p  +  x(A  +  2  WTT),  -  TT  <  ^  <TT, 

(a=2+y2) 

and  all  the  points  lie  on  the  spiral  r  =  p     x     •  e~y9/x. 

13.  Explain  the  fallacy  in  the  following  argument:  since 
^mni  __  e2nm  _  J?  where  m  and  «  are  any  integers,  therefore  rais 
ing  each  side  to  the  power  /  we  obtain  e~2mn  =  e~2nn. 


§57  a.    THE   FUNCTION  tan-i*  311 

14.  How  is  the  theory  of  logarithms,  as  laid  down  here,  har 
monized  with  the  elementary  notion  of  a  logarithm  such  as 

Iog10  ioo  =  2,  and  such  as    I    <//7  /  =  log  .#  ? 

We  may  define  w  =  loga  z  in  two  different  ways  :  (I)  we  may 
put  w  =  loga  z  if  the  pri)uipal  value  of  a"  is  equal  to  z  ;  (II)  we 
may  say  that  w  =  loga  z  if  any  value  of  aw  is  equal  to  z. 

Hence  if  a  =  e,  then  w  =  loge  z  according  to  the  first  defini 
tion  if  the  principal  value  of  ev  is  equal  to  z,  or  if  expw  =  z; 
and  thus  loge  z  is  identical  with  log  z.  But,  by  the  second 
definition,  w  =  loge  z  if 

ev  =  exp  (w  log  e)  =  z,    w  log  e  —  log  z, 

or  w  =  —  K-  ,  any  values  of  the  logarithms  being  taken.     Thus 
loge 

w  =  loge  z  =  L°gN+(Ams+2;/;7r)^ 

1+2   //7T/ 

so  that  w  is  a  doubly  infinitely  many-valued  function  of  z.     And 
generally,  according  to  this  definition,  logas  =  —  &•—  . 


15.    Show    that    log,  (i)  =  2  ;«TT//(I  +  2  «JT/),    log,  (—  i)  = 
(2  m-\-  I)TT//(I  +  2  ;z?r/),  ;;/  and  #  being  any  integers. 

§  67  a.   The  Function  t&n~lz 

In   §  56  we  defined  the   logarithmic  function  for  a  complex 
argument  by  the  integral 


We  now  introduce  a  new  variable  of  integration  Z  in  this  inte 
gral  by  the  following  substitution  already  investigated  in  §  15  ; 

(2\  »_i  +  t'Z    z       .1  -£     dt  _     2  idl 

\      '  »  »>  —  i     *»     •**   —   /  •~\o' 


312  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

The  integral  is  thus  transformed  into 

« 


the  new  upper  limit  is  connected  with  the  old  one  by  the  same 
equation  as  the  new  variable  of  integration  is  with  the  old  vari 

able  ;  that  is, 

/  N  i  -\-iZ      r,       .1  —  2 

(4)  z  =  —  -  —  ,    Z—t  -- 

i-t'Z  1  +  2 

But  when  an  integral  between  complex  limits  is  to  be  trans 
formed  by  the  introduction  of  a  new  variable  of  integration  we 
must  be  careful,  in  general  and  hence  here,  that  the  limits  of 
the  two  integrals  correspond  to  each  other  and  also  that  the 
paths  of  integration  correspond  —  at  least  whenever  we  are 
dealing  with  an  integral  which  is  not  completely  independent  of 
the  path.  The  logarithm  has  infinitely  many  values  according 
to  the  choice  of  the  path  of  integration  ;  we  have  arbitrarily 
chosen  one  of  these  values  as  principal  value.  We  shall  obtain 
the  best  notions  concerning  the  new  integral  by  using  the  prin 
cipal  value.  In  defining  the  principal  value  of  the  amplitude 
and  subsequently  the  principal  value  of  the  logarithm,  we 
drew  a  cut  in  §  54  along  the  half-axis  of  negative  reals  and  pro 
hibited  the  path  of  integration  from  crossing  this  cut.  It  is 
essential,  therefore,  to  determine  first  what  lines  of  the  Z-plane 
correspond  to  this  cut  in  the  s-plane.  From  the  results  of  §§  14 
and  15  we  know  already  that  a  straight  line  of  the  s-plane  cor 
responds  to  a  circle  or  again  to  a  straight  line  of  the  Z-plane  ; 
since  a  circle  is  completely  determined  by  three  of  its  points,  it 
will  only  be  necessary  to  find  the  points  of  the  Z-plane  corre 
sponding  to  three  points  of  this  cut.  As  in  §  15,  following  (3), 
we  have  the  following  pairs  of  corresponding  values  : 

z  =  o          —  i          oo 
Z=i          OQ         -  1  , 


§57  a.    THE   FUNCTION  tan'1*  313 

The  cut  in  the  .s-plane  thus  corresponds  in  the  Z-plane  to  that 
part  of  the  axis  of  pure  imaginaries  which  runs  from  Z  =  i 
through  infinity  to  Z  =  —  i.  We  obtain  accordingly  the  prin 
cipal  value  of  the  logarithm  when  we  so  choose  the  path  of 
integration  for  the  integral  (3)  that  it  does  not  cross  this  part  of 
the  Z-axis.  The  remaining  values  then  follow  from  this  princi 
pal  value  by  the  addition  of  arbitrary  integral  multiples  of  2  ?r/. 

Integral  (3)  without  the  factor  2  i  receives  a  specific  name 
based  upon  the  usual  terminology  in  the  theory  of  functions  of 
a  real  variable  ;  we  define  : 

I.    The  symbol  tan~!Z 

also  denotes  for  complex  values  of  Z,  any  one  of  the  values  of  the 
integral 


f 

Jo 


which  is  obtained  when  the  path  of  integration  is  chosen  arbitrarily 
(excepting  of  course  that  this  path  cannot  be  taken  through  one 
of  the  points  +  /  or  —  i,  since  this  symbol  of  integration  would 
then  have  no  meaning). 

II.  From  this  totality  of  values  we  then  select  as  the  principal 
value  that  otie  which  is  obtained  when  the  path  of  integration  is  not 
allou>ed  to  cross  the  cut  described  above. 

We  have  then  the  theorems  : 

III.  All  the  remaining  values  of  the  function  tan~lZ  are  obtained 
from  the  principal  value  of  this  function  by  the  addition  of  arbitrary 
integral  multiples  of  TT. 

Also  (on  account  of  Theorem  I,  §  57)  : 

IV.  The  principal  value  of  the  inverse  tangent  takes  on  each  com 
plex  value  w,  whose  real  part  u  satisfies  the  inequality 

(6)  ~-2<u^ 

at  one  and  only  one  point  of  the  plane  ; 


314  V.    MANY-VALUED  ANALYTIC   FUNCTIONS 

or  geometrically  : 

V.  The  Z-plane  cut  in  the  manner  specified  above   is  mapped 
conformally  by  the  principal  value  of  the  function  tan~lZ  upon  the 
parallel  strip  of  the  w-plane  bounded  by  the  straight  lines 

(7)  U=—7T/2,      U=+7T/2. 

In  this  transformation  the  parallels  to  the  z>-axis  correspond 
to  the  circles  through  the  two  points  Z  =  +  i  and  Z  =  —  i,  the 
parallels  to  the  ^-axis  to  the  circles  which  cut  those  through  / 
and  —  /  at  right  angles  ;  in  particular  the  &r-axis  corresponds  to 
the  Jf-axis,  the  z/-axis  to  that  part  of  the  F-axis  from  Z  =  —  i  to 
Z  =  +  i. 

We  get  likewise  from  the  corresponding  theorem  on  the 
logarithm : 

VI.  There  is  always  one  and  only  one  value  of  Z  for  which  one 
of  the  values  of  tan~l  Z  is  equal  to  an  arbitrary  preassigned  com 
plex  number  w. 

It  thus  follows,  as  in  §  57,  that  the  inverse  of  the  function 
w  =  tan"1  Z  is  a  function  of  w  which  is  single-valued  in  the 
whole  plane.  But  it  is  not  regular  in  the  whole  plane.  For,  by 
means  of  the  principal  value  of  the  function  tan"1  Z  a  point  of 
the  Z-plane  at  infinity  corresponds  to  a  point  of  the  w-plane  at  a 
finite  distance  from  the  origin,  viz.  to  the  point  w  =  7r/2.  Con 
versely,  for  the  inverse  function  not  a  finite  but  an  infinitely 
great  value  corresponds  to  the  point  w  =  IT/ 2  ;  and  the  same 
is  then  true  for  all  those  values  which  are  obtained  from 
w  =  7T/2  by  the  addition  of  integral  multiples  of  IT. 

To  investigate  the  behavior  of  the  inverse  function  in  the 
neighborhood  of  such  a  point,  we  use  the  process  of  inverting 
the  series.  Thus  let  us  put 

(8)  «,=tan-'Z=^ 


§57  a.    THE   FUNCTION  tan'1  z  315 

For  values  of  Z  sufficiently  large  we  can  develop  the  function 
under  the  sign  of  integration  in  a  series  of  decreasing  powers  of 
Z  and  then  integrate  ;  this  gives  :  * 


dZ, 


Z      3^3      s  Z' 

On  inverting  this  series  we  obtain  i/Z  represented  by  a  series 
of  powers  of  w  —  ir/2  with  positive  integral  exponents,  the  first 
term  being 

do)  _(„_!). 

By  division  of  series  (A.  A.  §  77)  we  obtain  then  a  development 

for  Z  in  powers  of  w with  increasing  integral  exponents ; 

the  first  term  is 


We  thus  have  the  theorem  : 

VII.    The  inverse  of  the  function  tan~^  Z  has  as  simple  poles 
the  point  w  =  -  and  the  points 


2 

where  k  is  an  arbitrary  integer  ;  at  each  of  these  points  its  residue 
is  equal  to  —  i  . 

*  We  notice  that  this  development  does  not  give  the  principal  value  of  tan-1  Z 
for  all  values  of  Z  for  which  it  converges,  but  for  only  those  values  whose  real  part 
is  positive. 


316  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

All  these  properties  of  the  function  inverse  to  tan"1  Z  belong 
also  to  the  function 

(13)  tan  w  =  coif-  —  w  j, 

as  is  easily  shown  from  the  properties  of  the  cotangent  function 
discussed  in  §  52.  As  a  matter  of  fact,  we  have  already  noticed 
in  §  53  that  the  integral  (5)  represents  the  inverse  of  the  function 
tan  w.  But  it  is  important,  especially  in  considering  compli 
cated  cases,  to  know  directly  that  all  solutions  of  the  equation 
tan  w  =  Z  can  be  represented  by  means  of  the  integral  (5)  when 
the  corresponding  path  of  integration  is  entirely  arbitrary. 
Moreover,  the  equation 

(14)  tan-1Z=  —  log  i-±^?  or  log  z  =  2  i  tan"1/^— ^\ 

21  I  —  iZ  \\  +  Z  J 

is  entirely  in  harmony  with  the  EULERIAN  relations  II,  §  40 ;  in 
fact,  if  we  put 

(15)  Z=tana' 
we  obtain : 

/  s\       i    i        i  -f-  iZ       i    i       cos  w  +  i  sin  w       i    ,        9,-«, 

(16)  —.  log  -!-—=—  log  -  !— ^ =  — .log^w  =  ze/ 

21  I   —  iZ         21  COS  W  —  I  Sin  W         21 

as  it  should  be. 

§  58.     The  Square  Root 

By  means  of  the  logarithmic  function  we  can  now  answer  the 
question  mentioned  at  the  end  of  the  first  chapter  about  the 
meaning  of  the  roots  of  complex  numbers ;  that  is,  about  the  in 
verse  of  the  function  zn  investigated  in  §  18.  To  be  sure, 
Theorems  III  and  IX  of  §  18  would  suffice  to  answer  this  ques 
tion  ;  but  we  notice  that  these  theorems  were  obtained  only  by 
representing  a  complex  number  in  terms  of  its  absolute  value  and 
amplitude,  and  this  is  equivalent  with  the  determination  of  the 
logarithm,  in  so  far  as  we  are  concerned  with  the  essential  point 


§58.    THE   SQUARE   ROOT  317 

of  the  question,  viz.  the  many-valuedness.  Of  course  we  can 
also  derive  those  theorems  purely  algebraically  if  we  assume  the 
fundamental  theorem  of  algebra  ;  but  this  is  much  less  direct.* 
In  general  an  algebraic  function  is  not  necessarily  simpler  in 
itself  than  a  transcendental  function.  With  the  aid  of  the  loga 
rithmic  and  exponential  functions,  we  now  study  in  this  para 
graph  the  function  "square  root"  as  the  simplest  example  of 
how  to  obtain  an  insight  into  the  nature  of  the  algebraic  depen 
dence  between  two  variables,  by  representing  both  of  them  as 
single-valued,  transcendental  functions  of  an  auxiliary  variable. 
Definition : 

I.    The  square  root  of  a  complex  number  z, 

(1)  s  =  Vz, 

is  a  complex  number  s  which  satisfies  the  equation : 

(2)  S"-  =  Z. 

If  we  introduce  an  auxiliary  variable  y  by  the  relation : 

(3)  *  =  <"> 

that  is,  if  we  put  rj  equal  to  one  of  the  values  of  the  function 
logs,  already  discussed  in  §  56,  s  is  also  expressible  as  a  single- 
valued  function  of  rj  as  follows.  Since  equation  (2)  is  to  be  pre 
served,  any  value  of  the  logarithm  of  one  side  must  be  equal  to 
a  value  of  the  logarithm  of  the  other  side  ;  hence  it  follows  that 

(4)  rj  =  one  of  the  values  of  log  (s2). 

But  these  values  separate  (VIII,  §  56)  into  two  classes :  the 
values  of  one  class  are  equal  to  2  log  s,  the  others  differ  from 
these  by  uneven  multiples  of  2  TTZ.  It  then  follows  that  every 

*  Cf.,  for  example,  H.  WEBER,  Lehrbuch  der  Algebra  (Braunschweig,  1895), 
Vol.  i,  page  107. 


3l8  V.    MANY- VALUED   ANALYTIC   FUNCTIONS 

value  of  log  s  must  be  representable  either  in  the  form : 

2  Vt>  ~°}         I?         2'  '" 

or  in  the  form  : 

-H 27T/,  k  =  0,    ±  I,    ±  2,  •••. 

Both  of  these  are  represented  in  the  one  form : 
-  +  /£TT/,  k=0,   ±  I,   ±  2,  ••• 

We  thus  obtain  the  following  result : 

II.  If  the  given  value  of  z  be  put  in  the  form  (3),  then  every 
value  of  s  belonging  to  it  is  representable  in  the  form 

(5)  *  =  >" 

in  which  k  is  any  integer. 

Conversely,  it  follows  from  the  equations  (u),  (12)  of  §40: 

III.  However  the  integer  k  in  (5)  may  be  chosen,  this  formula 
always  gives  a  value  of  s  which  satisfies  equation  (2),  and  there 
fore,  according  to  the  definition,  is  a  value  of  V ' z. 

This  is  also  expressible  in  another  manner.  We  understood 
77  above  to  be  a  definite  one  of  the  values  of  log  z ;  all  the  others 
are  then  of  the  form  77  +  2  kiri  where  k  is  an  integer.  We  there 
fore  obtain  all  the  values  (5)  directly,  if  77  in 

TJ 

is  now  understood  to  be  any  arbitrary,  not  a  fixed  value  of  log  z. 
Accordingly  we  may  state  theorems  II  and  III  as  follows : 


§59-    RIEM ANN'S   SURFACE  FOR  THE   SQUARE   ROOT      319 

IV.  We  obtain  all  the  pairs  of  corresponding  values  s,  z  which 
satisfy  equation  (2)  if  we  put 

(7)  j  =  ^,  *  =  ,i 
and  regard  rj  as  the  independent  variable. 

V.  If  we  take  the  principal  value  for  log  z  in  (6),  we  obtain  a 
definite  value  of  s  ;  we  call  it  the.  principal  value  of  tJie  square  root. 
Its  characteristic  property  is  tJiat  its  amplitude  \\i  satisfies  the  condi 
tions 

(8)  -f<*^ 

in  other  words,  tJiat  its  real  part  is  not  negative* 

Since  the  logarithm  is  an  infinitely  many-valued  function,  it 
might  appear  from  (6)  that  the  square  root  could  also  have  an 
infinite  number  of  values.  But  that  is  not  the  case.  All  the 
values  of  the  logarithm  follow  from  the  principal  value  by  the 
addition  of  2  kiri  where  k  is  an  arbitrary  integer.  If  this  integer 
be  even,  we  obtain  from  (6)  the  same  value  s  =  s0  as  when  the 
principal  value  of  the  logarithm  is  used ;  if  it  be  uneven,  we 
obtain  s=s0  •  <?""''=  —  ^0.  It  follows  accordingly  that  for  the  square 
root  there  is  only  one  value  beside  the  principal  value ;  or : 

VI.  To   every  value  (different  from  o  and  oo)  of  the  complex 
number  z,  there  belong  two  and  only  two  values  of  s  which  satisfy 
equation  (2). 

§  59.  The  RiEMANN'S  Surface  for  the  Square  Root 

In  order  to  make  the  square  root  a  single-valued  function  of 

position  on  a  surface,  we  do  not  need,  according  to   the  last 

theorem,   the    infinitely   many-sheeted   helicoid    surface    upon 

which  the  logarithm  is  represented,  but  it  is  sufficient  to  use  a 

*  If  the  real  part  of  the  square  root  is  zero,  then  the  positive  imaginary  value  is 
the  principal  value. 


320  V.    MANY- VALUED   ANALYTIC   FUNCTIONS 

two-sheeted  surface  arising  from  two  circuits  on  the  helicoid  (Fig. 
28).  In  this  connection  we  notice  that  the  "  second  value  " 
of  the  logarithm  (in  the  sense  of  definition  IV,  §  54)  again  fur 
nishes  the  principal  value  of  the  square  root.  Thus  at  the 
place  on  the  surface  of  the  logarithm  where  the  second  sheet 
is  joined  to  the  third,  the  first  sheet  in  the  surface  representing 
the  square  root  is  joined  to  the  second,  provided  that  to  every 
continuous  connection  between  the  values  of  the  function  there 


FIG.  28  FIG.  29 

shall  correspond  a  continuous  connection  between  the  parts  of 
the  surface.  But  this  cannot  be  represented  otherwise  in  space 
than  to  allow  the  generating  border  of  the  second  sheet  to  pierce  the 
part  of  the  surface  lying  under  it,  in  order  that  it  may  return  to 
its  initial'position  in  the  first  sheet  which  lies  under  the  second, 
thus  uniting  the  two  sheets  (Fig.  29). 

The  form  of  such  a  surface  is  most  easily  obtained  by  con 
structing  it  step  by  step.  Let  us  think  of  a  radius  in  the  plane 
unlimited  in  length  and  beginning  at  the  origin  which,  starting 
from  a  definite  initial  position  (say  <£  =  —  w),  turns  about  the  ori 
gin  in  the  positive  sense  and  by  such  a  movement  describes 
part  of  a  surface.  When  this  radius  has  returned  to  its  initial 
position  after  one  revolution,  the  surface  thus  generated  has 


§59-    RIEMANN'S   SURFACE   FOR  THE   SQUARE   ROOT      321 

two  borders  lying  adjacent  to  each  other.  But  these  are  not 
yet  to  be  united  with  each  other  ;  on  the  contrary,  let  the 
moving  boundary  be  pushed  beyond  the  fixed  one,  and  let  the 
turning  movement  continue  over  the  first  sheet  in  the  same 
sense  as  before  and  so  that  the  part  of  the  surface  thus  con 
tinually  generated  trails  behind  this  moving  radius.  When  this 
moving  boundary  completes  a  second  revolution,  it  is  allowed 
to  pierce  the  surface  lying  under  it  and  to  be  combined  with 
the  initial  boundary  lying  still  deeper. 

Figure  30  represents  a  section  of  the  surface  made  by  a  plane 
perpendicular  to  the  half-axis  of  negative  real  numbers.  It 

shows  how  the  left  part  of 

// 
the   first   sheet   is    bridged     ____ 

or    connected    along    this 

negative  axis  with  the  right 

part  of   the  second   sheet,  j  / 

and  how  the  right  part  of 

the  first  is  bridged  to  the  left  part  of  the  second. 

The  point  z  =  o,  about  which  the  sheets  are  regarded  as  hang 
ing  together  so  that  we  must  change  from  one  sheet  to  the 
other  in  making  a  circuit  about  this  point,  is  called  a  branch 
point  of  the  surface  (cf.  II,  §  55)  ;  it  is  in  fact  a  simple  branch 
point,  or  one  of  the  first  order.  In  the  same  way  the  point  oo  is  a 
simple  branch-point.  The  lines  along  which  the  two  sheets  pierce 
each  other  are  called  bridges'  (or  simply  cuts  or 


X 


*  In  Fig.  30  we  referred  to  the  sheets  of  the  surface  as  having  a  bridge  between 
them.  What  is  thus  called  (provisionally)  the  bridge  between  the  sheets  will  serve 
as  a  cut  in  the  s-plane  to  determine  two  branches  of  the  function  ;  in  this  case  the 
branches  are  assigned  to  the  upper  and  lower  sheets  respectively.  And,  con 
versely,  when  a  cut  has  been  employed  to  locate  branches,  it  is  often  convenient 
to  use  that  cut  as  a  bridge  on  the  RIEMANN'S  surface  and  to  call  it  a  branch-cut. 
Thus  in  the  case  above  when  V0  and  —  Vz  are  the  two  branches  the  axis  of  nega 
tive  real  numbers  is  a  branch-cut  for  the  corresponding  RIEMANN'S  surface. 
—  S.E.R. 


322  V.    MANY-VALUED  ANALYTIC   FUNCTIONS 

Therefore  upon  this  surface  the  square  root  is  a  single-valued 
function  of  position ;  not  only  a  definite  value  of  z  but  also  a 
definite  value  of  s  =  ~\/z  is  assigned  to  each  of  its  points. 
Hence  s  is  also  a  continuous  function  of  position  on  this  surface  ; 
if  a  point,  progressing  continuously,  takes  on  all  the  values  on 
a  closed  curve  on  the  surface  itself  (not  merely  on  its  projection 
on  the  2-plane),  then  the  corresponding  values  of  the  square 
root  also  change  continuously.  Conversely,  only  ONE  point  of 
the  surface  corresponds  to  each  pair  of  values  (z,  s),  which  satisfies 
equation  (2),  §^58.  In  order  that  this  may  be  true  without  excep 
tion  we  stipulate  further ;  the  branch-point  is  counted  as  only  one 
point  of  the  surface  corresponding  to  the  pair  of  values  (o,  o). 
But  every  other  point  of  the  bridge  or  branch-cut  represents  two  points 
of  the  surface,  one  of  which  belongs  to  one  part,  the  other  to  the  other 
part,  of  the  surface  divided  at  this  cut. 

It  is  important  that  we  have  a  clear  notion  of  what  is  essen 
tial  and  what  is  not  essential  in  this  geometrical  representation 
of  the  connection  between  the  values  of  the  function  by  means 
of  the  RIEMANN'S  surface.  The  branch-points  z  =  o  and  z=  oo 
are  essential ;  to  change  them  would  mean  to  change  the  func 
tion  s  =  -Vz  to  some  other  function,  not  merely  to  give  another 
form  to  the  geometrical  picture.  On  the  other  hand,  the  form 
of  the  branch-cut  is  entirely  unessential ;  it  must  only  connect 
the  points  o  and  oo.  That  it  coincides  with  the  axis  of  negative 
real  numbers  is  only  a  consequence  of  the  manner  in  which  we 
defined  the  principal  value  of  the  amplitude,  and  thereby  of  the 
logarithm  in  I,  §  54.  We  might  make  some  other  arbitrary 
assumption  in  order  to  define  a  first  sheet  of  our  surface.  Such 
an  assumption  is  formulated  geometrically  as  follows :  Let  us 
draw  a  definite  line  from  o  to  oo  not  intersecting  itself;  then 
let  us  choose  for  a  definite  point  z0,  not  lying  upon  this  line,  one 
of  the  two  values  belonging  to  j,  say  J0,  and  take  at  any  other 


§  59-    RI  EM  ANN'S   SURFACE  FOR  THE   SQUARE   ROOT      323 

point  Si  that  value  for  s  which  is  obtained  when  a  z-path  is 
drawn  from  ZQ  not  intersecting  the  line  and  when  in  this  way 
s  changes  continuously  from  SQ.  Let  us  then  take  two  such 
sheets  and  connect  them  crosswise  along  the  cut.  We  thus 
obtain  a  surface  upon  which  V ' z  is  a  single-valued  and  continu 
ous  function  of  position. 

If  we  wish  to  take  into  account  in  this  geometrical  representa 
tion  the  arbitrary  manner  of  choosing  the  cut,  we  must  regard 
the  sheets  as  mwable  over  each  other  in  such  a  way  that  tfie  cut 
can  be  shifted  without  breaking  the  connectivity.  To  be  sure  this 
supposes  that  the  one  sheet  is  partially  shoved  through  the  other 
without  tearing  them  (that  is,  if  the  old  cut  be  Fand  the  new 
one  be  V* ,  the  part  of  the  lower  sheet  between  ^and  V  be 
comes  part  of  the  upper  sheet,  and  vice  versa) ;  *  but  there  is  no 
necessity  whatever  of  ascribing  the  property  of  impenetrability 
to  the  sheets,  since  they  are  only  geometrical  and  not  physical 
creations.  In  general,  this  cut  is  only  a  necessary  makeshift ; 
a  continuous  transition  from  one  value  of  the  function  to  the 
other  belonging  to  the  same  value  of  the  argument,  does  not 
take  place  at  the  cut  just  as  it  does  not  at  other  places  on  the 
surface  (with  the  exception  of  the  branch-points).  In  the  appli 
cation  of  this  idea  it  is  convenient  to  make  the  following  stipu 
lations  —  and,  in  fact,  once  for  all,  since  we  shall  frequently  be 
concerned  with  similar  relations  : 

//  is  assumed  that  there  is  no  connection  along  a  lint  between  two 
parts  of  a  surface  which  is  divided  by  such  a  li)ie.  A  point  which 
moves  upon  a  surface  of  this  kind  must,  when  it  comes  to  such  a 
line  (or  cut),  never  cross  the  cut  to  the  otJier part  of  the  surface. 

(In  Fig.  30  the  left  half  of  the  lower  and  the  right  half  of  the  upper  "  sheet" 
represents  the  one  part,  the  right  half  of  the  lower  and  the  left  half  of  the 
upper  represent  the  other  one  of  the  two  "parts  of  the  surface,1'  mention  of 
which  has  just  been  made  in  the  above  statement.) 

*  The  sentence  in  parenthesis  inserted  by  the  translator. 


324  V.    MANY-  VALUED   ANALYTIC   FUNCTIONS 

EXAMPLES 

1.    For  the  function  s  =  Vz  put  z  —  r(cos  <£  +  /sin  </>);  that  is, 
s  —  vV(  cos  —  +  /sin*  )  (where  r  may  be  put  equal  to  i)  and 

construct  a  table  of  corresponding  values  of  <j>,  z,  and  s  using  for 
this  purpose  <j>  =  o,  -,  TT,  etc.  ;  show  in  this  way  that  the  values 

2 

of  s  will  not  repeat  until  </>  makes  two  circuits  about  the  origin. 

That  z  =  o  is  here  a  branch-point  is  shown  by  describing 
closed  paths  around  it.  (Cf.  footnote  following  II,  §  55.)  Let 
the  variable  start  from  the  point  z=i  and  describe  a  circle 
about  the  origin  ;  let  the  function  s  =  V  '  z  start  from  the  point 
z  =  i  with  the  value  s  =  +  i  and  thus  r  =  i  ,  </>  =  o.  As  z  now 
describes  a  circle  in  the  positive  direction,  r  remains  =  i  and 
<j>  increases  from  o  to  2  w.  When  the  variable  has  returned  to 
*  =  +i,wehave  z  =  COS2v  +  jsin2Vt 
and  hence  s  =  V  'z  =  cos  ir-\-i  sin  TT  =  —  i  ; 

the  function  has  now  not  the  original  value  +  i,  but  the  other 
value  —  i.  The  same  thing  takes  place  when  the  variable  start 
ing  from  z  =  i  describes  any  other  closed  path  around  the  origin, 
since  this  path  can  be  gradually  deformed  into  the  circle  with 
out  thereby  passing  through  the  origin. 

And,  in  general,  if  s  start  with  the  value  s0  at  any  point  z0  at 

which  *b  =  r0  (cos  <fo  +  ''  sin  <fo) 


SQ  =  r    (cos  i  <£0  +  /  sin 

and  if  z  describe  a  closed  path  around  the  origin  once  in  the 
positive  direction,  then,  on  returning  to  ZQ,  we  have 


and  hence  s  =  r01/2[cos  Q  <£0  +  T)  4-  /sin  (|  <£0  +  TT)] 


§  59-    RIEM ANN'S   SURFACE   FOR  THE   SQUARE   ROOT      325 


If  the  variable  describe  the  closed  path  twice,  or  any  other 
closed  path  around  the  origin  twice,  then  the  amplitude  of  z 
increases  by  4  IT,  that  of  s  by  2  TT,  and  hence  the  function  again 
takes  on  its  original  value. 

2.  Show  by  means  of  the  transformation 

""  t 
that  z=  x,  t=o  is  a  branch-point  for  s=  V ' z. 

3.  A  case  very  similar  to  that  of  Ex.  i  is  the  function 

,=(«-. )Vi 

Here  2=0,  but  not  0=1 
is  a  branch-point.  For, 
let  us  consider  the  point 
z  =  i  for  which  s  =  o, 
and  let  z  describe 
around  it  a  circle  with 
radius  r,  starting  at 
c  —  i  -f  r  on  the  real 
axis  (cf.  Fig.).  If  we 


put 


i  =  r  (cos  <£  -f-  /  sin 


then         s  =  r  (cos  <£  +  /  sin  <£)  Vi  4-  r  cos  <£  -f  r/  sin  <£. 

As  ;-  remains  constant  and  <f>  increases  from  o  to  2  TT  the  factor 
r(cos  <f>  -|-  /  sin  <£)  does  not  change  its  value.  To  study  the 
behavior  of  the  second  factor,  let  us  put 

i  +  r  cos  <£  =  p  cos  \l/ 
r  sin  <£  =  p  sin  ^  ; 

thus  p  is  the  straight  line  oz  and  «/r  the  angle  it  makes  with  the 
real  axis,  and  we  have 


s  =  r(cos 


sn 


sn  4- 


326  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

Therefore,  if  the  circle  does  not  inclose  the  origin,  \{/  passes 
through  a  series  of  values  beginning  with  o  and  ending  with  o, 
and  hence  s  does  not  change  its  value.  But  if  the  circle  be  so 
large  that  the  origin  also  lies  within  it,  ij/  increases  from  o  to  2  ?r, 
and  hence  in  this  case  the  original  value  s  =  rp^  passes  into  —  rp1. 
We  thus  confirm  the  statement  that  only  the  point  z  =  o  and  not 
the  point  z  =  i  is  a  branch-point. 

4.  It  is  sometimes  desirable  to  consider  the  function  (z—  i)^/z 
of  the  previous  example  as  derived  from 


by  making  a  =  i.  A  line  inclosing  the  point  z=i  can  then 
be  regarded  as  having  at  first  inclosed  the  two  points  z=  i  and 
z  =  a  which  were  subsequently  made  to  coincide.  Now  z  =  i , 
z  =  a,  and  z  =  o  are  all  branch-points  of  the  function  s'.  A 
closed  path  which,  starting  from  ZQ,  makes  a.  circuit  around  both 
points  i  and  a,  can  be  replaced  by  closed  paths,  each  of  which 
incloses  only  one  of  these  points.  And  if  s'  start  from  z0  with 
the  value  j'0,  on  encircling  the  point  a  it  passes  into  —  s'Q,  and 
then  on  encircling  the  point  i ,  —  s'Q  passes  into  s'0  again.  The 
function  returns  therefore  to  z0  with  its  original  value.  This  is 
true  as  a  approaches  the  point  i,  and  when  these  branch-points 
coincide  the  common  point  obviously  ceases  to  be  a  branch-point. 

5.    Discuss  next  the  function 

Vz  —  a          ,  , 

— ,   a,b  complex. 
z  —  b 

Here  z  —  a  and  z  =  b  are  both  branch-points.     For,  if  we  first 
let  z  describe  a  closed  path  around  the  point  a  starting  from 
any  point  z0  say,  but  not  inclosing  the  point  b,  and  if  we  accord 
ingly  put          z-a  =  r  (cos  j>  +  i  sin  <f>) , 
z0  —  a  =  r0(cos  <£0  +  i  sin  </>0), 


§  59-    RIEM  ANN'S   SURFACE   FOR  THE  SQUARE   ROOT      327 
then  the  initial  value  of  s,  denoted  here  by  j1?  is 


sn 


4-  TO(COS  <£<)  +  i  sm 

After  the  closed  path  is  described  once  in  the  positive  direc 
tion,  (fr0  has  increased  by  2  TT  and  hence  the  resulting  value  of  j, 
denoted  here  by  s2,  is 

j  _  r^  [cos  (i  <fr  o+ 1 »0  +  i  sm(i  <fto  + 1  w)] 
\a  —  b  +  r0(cos  (fr0  +  i  sin  <fr0)]* 

Here  the  denominator,  and  therefore  the  quantity  $Jz  —  b  can 
not  have  changed  its  value  because  for  it  z  =  b  and  not  z  =  a  is 
a  branch-point ;  z  has  thus  described  a  closed  path  which  does 
not  include  the  branch-point  of  this  expression.  Let 


2 

be  a  root  of  the  equation  a3  =  i  ;  then,  since 

cos  (-g-  (fr0  -f-  |  TT)  -f-  /  sin  (-3-  (fr0  -{-  |-  TT)  =  (cos  \  (fr0  +  /  sin 

(cos  -|  TT  4- '  sin  J  ?r), 
we  can  write  s2  =  ft^. 

Now  let  z  describe  a  second  closed  path  around  the  point  a ; 
then  s  starts  at  z0  with  the  value  s2  =  as^  and  acquires  after  com 
pleting  the  circuit  the  value 

s3  =  as2  =  a2^. 

After  a  third  circuit  s  acquires  the  value  ofsl'1  that  is,  the 
original  value  s1  since  «3=i.  If  we  had  started  from  ZQ  with 
the  value  s2  instead  of  jj,  we  should  have  obtained  s3  and  ^  after 
one  and  two  circuits  respectively;  and  if  s3  had  been  the 
original  value,  it  would  have  changed  into  sl  and  s2  successively. 

Show  now  that  similar  results  are  obtained  when  z  is  made 
to  describe  a  closed  path  including  only  the  point  b  :  and  further 


328  V.    MANY- VALUED   ANALYTIC   FUNCTIONS 

that  repeated  circuits   around  a  branch-point  interchange  the 
function-values  in  cyclical  order. 

Discuss  also  what  takes  place  when  z  describes  a  closed  path 
including  both  points  a  and  b. 

§  60.   Connectivity  of  this  Surface 

One  frequently  encounters  the  problem  to  apply  the  general 
theorems  of  Chapter  IV  concerning  single-valued  functions  of  z 
to  such  functions  which  are  single-valued  functions  of  position 
on  any  RIEMANN'S  surface  other  than  the  plane,  or  the  sphere. 
Now  those  theorems  depend  upon  the  fundamental  theorem  of 
integration  in  §  36  due  to  CAUCHY,  and  this  again  depends 
upon  the  substitution  of  an  integral  taken  along  a  closed  curve 
for  a  sum  of  integrals  taken  around  sufficiently  small  regions  of 
the  surface.  If,  therefore,  these  theorems  are  to  be  applied  to 
any  other  surface,  we  must  first  determine  whether  any  closed 
curve  on  this  surface  also  completely  bounds  a  region  of  the 
surface  ;  we  shall  see  that  this  is  by  no  means  the  case  for  all 
surfaces. 

The  problem  is,  as  we  see,  a  qualitative  one ;  it  has  nothing 
to  do  with  comparing  the  dimensions  of  the  surfaces,  but  is 
to  be  answered  in  the  same  manner  for  all  surfaces  which 
can  be  transformed  into  each  other  by  continuous  deformation 
(stretching  and  bending)  without  tearing.  It  thus  belongs  to  a 
chapter  in  geometry  which  is  customarily  called  analysis  situs 
or  topology,  and  which  in  general  treats  of  those  properties  of 
geometrical  forms  common  to  all  forms  which  can  be  trans 
formed  into  each  other  by  stretching  and  bending  without 
tearing.  Moreover,  in  the  treatment  of  this  question  the  geo 
metrical  forms  may  be  supposed  to  be  penetrable  or  to  be 
impenetrable.  But  according  to  previous  assumptions  it  is 
quite  necessary  for  our  purpose  to  regard  them  as  impene- 


§  6o.    CONNECTIVITY  OF  RIEM ANN'S  SURFACE  FOR    Vz      329 


trable.     We  can  then  deform  *  our  surface  into  a  sphere  in  the 
following  manner : 

Let  us  first  draw  out  the    inner  sheet  further   through   the 
branch-cut  (Fig.  31   a).     This   process   is  continued  until   the 


FIG.  31 

entire  inner  spherical  sac  is  drawn  out ;  a  sharp  edge  (&}  now 
appears  at  the  place  at  which  the  cut  had  been  made.  Let  us 
next  smooth  this  off  (c]  and  the  sphere  (</)  is  the  final  result. 

We  can  also  arrange  this  deformation  process  somewhat  dif 
ferently.  We  can  think  of  the  inner  sphere  as  flattened  out 
more  and  more  until  it  finally  becomes  a  doubly  covered  circular 
flat  disc.  It  is  then  evident  that,  by  pulling  the  two  sheets  of 
this  disc  through  each  other,  a  sphere  with  a  pocket  sunk  in  it 
results  (Fig.  32  b).  If  this  pocket  be 
gradually  flattened  out,  we  obtain  finally 
a  sphere. 

The  question  as  to  the  possibility  of 
a  continuous  deformation  of  one  surface 
into  another  need  not  be  emphasized 
here.  After  all,  two  surfaces  are  equivalent  for  the  present 
investigation  merely  when  they  are  so  related  that  a  continuous 
path  on  one  surface  corresponds  in  the  deformation  to  a  contin 
uous  path  on  the  other.  For  then  every  closed  line  on  the  one 
surface  which  completely  bounds  a  part  of  the  surface,  corre- 

*  A  large  number  of  figures  explaining  such  processes  of  deformation  are  to  be 
found  in  the  work  by  FR.  HOFMANN,  Methodik  der  stetigen  Deformation  zweiblatt- 
riger  RlEMANNSCHER  Flache,  Halle,  1888. 


FIG.  32 


330 


V.    MANY-VALUED   ANALYTIC   FUNCTIONS 


spends  on  the  other  to  a  closed  line  with  the  same  property 
(otherwise  a  continuous  path  on  the  one  surface,  which  connects 
two  points  on  opposite  sides  of  this  closed  line,  could  not  corre 
spond  to  a  continuous  path  on  the  second  surface,  contrary  to 
the  hypothesis).  But  such  a  correspondence  between  two  sur 
faces  is  obtained  also  as  follows  : 

Let  us  divide  the  given  surface  into  any  number  of  parts,  tak 
ing  care  that  we  know  in  what  manner  they  are  connected  at 
the  new  borders.  We  then  deform  each  of  the  parts  so  obtained 
without  tearing  and  without  uniting  the  parts  just  divided. 
Next  lay  the  deformed  parts  side  by  side  so  that  they  will  join 
in  pairs  with  such  parts  of  their  borders  as  originally  belonged 
together  ;  and  finally  unite  these  borders. 

In  the  case  under  discussion  the  deformation  takes  place  as 
follows  :  Mark  the  right  bank  (that  is,  the  one  lying  on  the  side 
of  positive  y)  of  the  branch-cut  in  each  sheet  by  hatching 


y////////////// 
A 


(Fig.  33,  A).  Then  make  an  incision  along  the  branch-cut 
through  both  sheets,  each  sheet  thus  appearing  as  a  sphere,  or 
as  a  plane,  respectively,  with  a  cut  —  the  latter  representation 
being  the  more  convenient  here.  By  turning  the  two  sides  of 
the  cut  apart  around  the  origin  in  opposite  directions,  we  contract 
the  surface ;  continue  this  way  until  the  angle  at  the  origin, 
which  is  2  TT,  is  reduced  to  TT.  Proceed  the  same  way  with  the 


§6oa.    RATIONAL   FUNCTIONS   OF  z   AND   s  =  Vz         331 

other  sheet.  When  both  sheets  are  deformed  in  this  way,  place 
them  in  the  plane  close  together  and  in  such  a  way  that  the 
smooth  bank  of  the  cut  in  one  sheet  lies  adjacent  to  the  hatched 
bank  of  the  cut  in  the  other  sheet,  just  as  they  were  originally. 
(It  can  also  be  so  arranged  by  suitably  stretching  the  banks  that 
the  points  of  the  banks  that  were  originally  side  by  side  are 
exactly  so  placed  after  deformation.)  Finally  let  us  unite  these 
banks.  We  obtain  in  this  way  a  smooth  plane,  or  a  sphere,  re 
spectively  (Fig.  33  <•). 

In  accordance  with  all  these  methods  of  deformation,  we 
therefore  obtain  the  theorem  : 

The  tiuo- sheeted  Ri  EM  ANN'S  surface  with  tivo  branch-points  has 
the  same  connectivity  as  tJie  sphere. 

§  60  a.     Rational  Functions  of  z  and  s  =  Vz 

In  the  investigation  of  any  algebraic  function  of  z — called  s 
for  example  —  it  is  appropriate  to  consider  at  the  same  time  all 
the  functions  which  can  be  expressed  rationally  in  terms  of  z 
and  s.  Every  such  function  has  only  one  definite  value  at  any 
point  of  the  RIEM ANN'S  surface  on  which  s  is  single-valued  ;  this 
value  is  obtained  by  giving  to  s  in  the  corresponding  expression 
exactly  that  value  which  belongs  to  this  point  of  the  surface. 

For  the  case  s  =  ^/z  then,  z  =  ^becomes  a  single-valued  func 
tion  of  s;  we  can  therefore  transform  at  once  every  rational 
function  of  s  and  z  into  a  rational  function  of  the  one  variable 
s.  But  when  a  complicated  algebraic  relation  exists  between  z 
and  s,  it  is  not  in  general  possible,  the  proof  of  which  will  not 
be  given  here,  to  introduce  an  auxiliary  variable,  by  which  z 
and  s  can  both  be  represented  as  rational  functions.  We  shall 
therefore  make  no  use  of  the  possibility  of  such  reduction  in 
the  above  simple  case,  but  shall  investigate  the  rational  functions 
of  s  and  z  directly  as  such. 


332  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

We  can  reduce  every  such  function  to  a  certain  simple  normal 
form.  We  can  represent  it  for  the  present  as  the  quotient  of  two 
rational  integral  functions  of  s  and  z,  and  then  remove  all  higher 
powers  of  s  occurring  in  numerator  and  denominator  by  means  of 
the  equations  : 

(l)  S*  =  Z,  S*  =  SZ,  S*  =  Z\  £  =  SZ\—\ 

in  this  way,  the  fraction  reduces  to  the  form 


in  which  the  ^'s  are  rational  integral  functions  of  z  alone.  Mul 
tiplying  numerator  and  denominator  by  gs(z)  —  sg4(z)  gives  the 
form  : 


Sf-Xt 

or 

(4)  r,(z)  +  sr,(z) 

in  which  r±  and  r2  are  rational  (fractional)  functions  of  z  alone. 
Therefore  every  rational  function  of  z  and  s  given  above  may  be  put 
in  this  form. 

Common  zeros  of  numerator  and  denominator  can  eventually 
be  removed  by  this  arrangement  or  new  ones  could  be  intro 
duced  ;  this  is  to  be  treated  as  in  §  20. 

To  express  the  variable  a-  as  a  rational  function  of  z  and  s,  we 
write  : 

(5)  *  =  *(*,  *). 

Then  the  value  of  the  function  jR(z,  s)  belongs  to  one  of  the  two 
points  which  lie  on  the  RIEMANN'S  surface  over  a  point  z  of  the 
plane,  and  the  value  of  the  function  R(z,  —  s)  belongs  to  the 
other  one  of  the  two  points,  provided  that  a  definite  one  of  the 
two  values  of  V^  corresponding  to  a  given  z  is  denoted  by  s. 


§  6i.  THEOREMS  AND  RIEM ANN'S  SURFACE  FOR  V~      333 

However,  this  symbolism  is  somewhat  tedious,  and  on  this  account 
we  frequently  write  simply  o-=/(z),  and  agree  that  the  symbol  z  is 
always  to  designate  a  definite  point  of  the  surface,  no  matter  which 
of  the  two  points  it  is  that  corresponds  to  the  same  value  of  the 
complex  variable  s.  The  other  one  could  then  be  designated  by 
z  say  (different  from  the  meaning  of  this  symbol  as  used  in  §  n). 
Conversely,  if  a  function  of  a  complex  variable  z  be  so  de 
fined  that  the  two  values  of  this  function  belong  to  each  value 
of  z,  and  that  these  values  are  so  arranged  on  the  two  sheets  of 
our  two-sheeted  RIEMANN'S  surface  that  only  one  of  these  values 
belongs  to  each  point  of  the  surface,  and  that  in  this  way  values 
of  the  function  differing  by  an  indefinitely  small  amount  corre 
spond  to  points  of  the  surface  indefinitely  near  each  other,  then 
we  call  such  a  function  single-valued  on  the  RIEMANN'S  surface. 
But  not  every  function  single-valued  on  our  RIEMANN'S  surface 
is  a  rational  function  of  s  and  z ;  this  is  as  improbable  as  that 
every  single-valued  function  of  z  alone  is  a  rational  function  of 
z.  In  §  44  we  became  acquainted  with  functions  of  z  alone  by 
means  of  which  we  could  determine  whether  or  not  a  given 
function  is  a  rational  function  of  z :  we  could  draw  conclusions 
about  the  nature  of  the  function  in  general  from  its  behavior  in 
the  neighborhood  of  any  individual  point.  This  was  possible 
on  account  of  the  fundamental  theorems  on  integration  due  to 
CAUCHY  ;  to  obtain  corresponding  theorems  for  the  functions 
on  a  RIEMANN'S  surface,  we  must  apply  those  theorems  of 
CAUCHY  to  functions  which  are  first  defined  to  be  single-valued, 
not  in  the  s-plane  but  on  such  a  surface. 

§  61.     Application  of   CAUCHY'S  Theorems  to  Functions  which 
are  single-valued  on  the  RIEMANN'S  Surface  for  v* 

To  properly  attack   this    problem  we  must  be    clear   at   the 
start  as  to  the  meaning  of  such  terms  as  regular,  pole,  essential 


334  v-    MANY- VALUED   ANALYTIC   FUNCTIONS 

singularity  when  applied  to  this  surface ;  this  is  necessary,  since 
the  former  definitions  of  these  terms  apply  only  to  functions 
which  are  single-valued  on  the  s-plane  itself. 

No  difficulty  whatever  presents  itself  in  a  domain  of  the  sur 
face  which  contains  no  branch-point.  Every  such  Domain  can 
be  constructed  from  parts  of  the  surface  each  of  which  lies 
entirely  in  one  sheet  of  the  surface ;  we  can  then  apply  the 
former  definitions  and  theorems  directly  to  each  such  part  of 
the  surface. 

It  is  different  in  the  neighborhood  of  a  branch-point :  the 
former  definitions  do  not  apply  to  such  a  point.  But  we  can 
map  the  neighborhood  of  the  branch-point  reversely  and  uniquely 
upon  the  neighborhood  of  the  origin  of  an  auxiliary  plane  by 
the  substitution  * 
(i)  z=P,dz  =  2tdt, 

and  then  study  in  this  plane  all  the  functions  to  be  investigated. 
It  is  therefore  essential  to  so  determine  all  definitions  that  they 
depend  upon  the  former  definitions  for  their  meaning  in  the 
auxiliary  plane.  Accordingly,  we  define  : 

I.  A  function  f(z)  of  z  is  called  "  regular  on  the  RlEMANKTS 
surface  "  in  the  neighborhood  of  the  branch-point  o,  when  it  is  trans 
formed  by  the  substitution  (/)  into  a  function  <j>  (/)  of  the  auxiliary 
variable  t,  which  is  regular  in  the  neighborhood  of  the  origin  of  the 
t-plane  in  the  sense  of  the  former  definition. 

*  Since  the  inverse  of  the  function  j=Vs,  by  which  we  have  defined  this 
RlEMANN'S  surface,  is  a  single-valued  function  of  s,  we  could  use  this  s  itself  as  an 
auxiliary  variable  here  and  thus  obtain  a  single-valued  representation  on  the  /-plane 
not  only  of  the  neighborhood  of  the  branch-point  but  also  of  the  entire  surface. 
But  since  it  is  not  in  general  possible,  as  we  have  seen  in  §  60  a,  to  find  an  auxiliary 
variable  having  this  property  for  complicated  algebraic  functions,  we  must  be  satis 
fied  with  mapping  the  neighborhood  of  the  branch-point  and  then  use  a  particular 
auxiliary  variable  for  each  branch-point. 


§  6i.   THEOREMS  AND  RIEMANN'S  SURFACE  FOR  vG     335 

In  this  connection  it  is  to  be  noticed  that  it  is  not  true  that  a 

• 

function,  regular  only  upon  the  surface  and  not  at  the  same  time 
in  the  s-plane,  has  everywhere  a  definite,  finite  derivative  with 
respect  to  z.  An  example  is  the  function  s  =  V^ ;  its  derivative  : 

W  7  =  —^ 

dz      2^/z 
is  not  finite  for  z  =  o. 

If,  in  the  further  study  with  substitution  (i),  we  obtain  a 
function  of  /which  is  regular  at  all  points  of  a  certain  neighbor 
hood  of  the  origin,  this  point  itself  excepted,  we  define  : 

II.  According  as  this  function  of  t  has  a  pole  (non-essential 
singularity)  or  an  essential  singularity  at  the  point  t  =  O,  we  say 
that  t/ie  branch-point  is  a  pole  or  an  essential  singularity  for  the 
assigned  function. 

And  further : 

III.  In  tJie  case  of  a  pole  at  tJie  branch-point,  the  order  of  the 
infinity  of  the  function  is  to  be  deter  mined  from  t  and  not  from  z  : 
thus,  for  example,  the  function  i/z  considered  as  a  function  of 
the  surface  has  a  pole  of  the  second  order  at  the  origin. 

Corresponding  to  this  we  say  of  a  function  which  is  regular 
at  a  branch-point,  that  it  has  a  zero  of  the  ;;/th  order  at  this 
branch-point  when  the  function  into  which  it  is  transformed  by 
substitution  (i)  has  a  zero  of  the  ;;/th  order  at  the  origin.  This 
is  also  expressed  as  follows : 

IV.  In  the  neighborhood  of  a  branch-point  we  consider  ~\fz,  not 
z,  as  an  infinitesimal  of  the  first  order. 


In  accordance  with  the  terminology  thus  defined,  we  state  the 
[lowing  theorem  : 
V.    The  integral 


(3)  J/M* 


336      V.  MANY- VALUED  ANALYTIC  FUNCTIONS 

taken  along  any  curve  which  completely  boiinds  a  part  of  the  sur*. 
face,  is  equal  to  zero  when  the  function  f(z]  is  regular  over  this  part. 

For,  if  this  part  contains  no  branch-point  within  it,  we  can 
divide  it  into  a  number  of  pieces  each  of  which  lies  entirely  in 
one  sheet  of  the  surface.  For  each  such  piece  the  earlier  proof 
is  then  valid;  and  if  we  subsequently  unite  these  pieces,  the 
integrals  taken  along  the  lines  between  the  pieces  drop  out  as 
in  §  29  (Fig.  15),  and  only  the  integral  taken  along  the  given 
curve  remains. 

But  when  the  integral  is  to  be  taken  along  a  curve  which  in 
closes  the  branch-point  at  the  origin,  we  map  the  neighborhood 
of  the  branch-point  on  the  neighborhood  of  the  origin  of  the 
/-plane  by  substitution  (i) ;  integral  (3)  is  thus  transformed  into 

2  \$(f)tdt.  A  curve  which  completely  incloses  this  branch 
point  on  the  surface,  is  projected  on  the  s-plane  into  a  curve 
which  there  encircles  the  branch-point  twice ;  this  curve  is 
mapped  in  the  /-plane  into  a  curve  making  just  one  circuit 
about  the  origin  ;  according  to  hypothesis  the  function  <£(/),  as 
also  the  function  /</>(/)  is  regular  inside  of  this  curve;  the 
integral  is  therefore  zero  in  the  /-plane,  and  this  result  is  appli 
cable  to  the  given  integral  on  the  surface. 

We  treat  the  neighborhood  of  the  branch-point  at  infinity  in 
a  similar  manner  with  the  aid  of  the  substitution  z  =  /~2. 

Finally,  if  we  are  considering  a  domain  which  contains  one 
or  both  branch-points  in  its  interior,  we  separate  it  in  the  neigh 
borhood  of  the  branch-points  into  pieces  each  of  which  lies  en 
tirely  in  one  sheet  of  the  surface  ;  then  the  theorem  holds  for 
each  of  these  separate  pieces,  and  on  combining  the  integrals 
those  taken  along  the  paths  between  the  pieces  again  disappear. 

Moreover,  in  passing  from  CAUCHY'S  theorem  on  integration 
to  the  expansion  in  series  according  to  the  CAUCHY-TAYLOR 


§  6i.   THEOREMS  AND  RIEMANN'S  SURFACE  FOR 


337 


theorem,  we  encounter  no  difficulty  whatever  if  we  remain 
away  from  the  branch-points.  In  particular,  when  a  is  a  value 
different  from  o  and  oo,  the  binomial  expansion 


I 

converges,  providing  z  remains  inside  of   a  circle  which  goes 
through  the  branch-point ;  or,  analytically,  if  (cf.  Fig.  34) 

(5)  \z-a\<\a\ 

(cf.  the  corresponding  theorem  for 
real  variables,  A.  A.  §  70).  In  fact, 
this  expansion  gives  the  one  or  the 
other  branch  of  the  function  accord 
ing  as  the  factor  Va  standing  in 
front  of  the  brackets  takes  the  one 
or  the  other  value  (only  single-valued 
functions  of  z  itself  are  inside  of 
the  brackets).* 

But  to  apply  the  former  conclusions  to  the  integral 


(6) 


—•f 

2   TTlJ 


*-(' 


taken  along  a  circle  of  radius  r  encircling  the  branch-point 
twice,  understanding  that  £  is  here  a  quantity  whose  absolute 
value  is  smaller  than  r,  we  must  observe  that  inside  of  the  do 
main  which  is  bounded  by  the  curve  of  integration,  the  function 
to  be  integrated  now  becomes  infinite  not  in  one  point  but  in 
two ;  viz.,  in  the  two  points  £  and  £  of  the  surface  which  lie  one 

*  We  are  not  to  conclude  that  series  (4)  must  always  furnish  the  principal  value 
of  z  when  we  use  the  principal  value  of  Va-  That  is  the  case  only  when  the 
straight  line  connecting  a  and  z  does  not  cross  the  half-axis  of  negative  real 
numbers. 


338  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

above  the  other  and  belong  to  the  same  value  of  the  argument. 
Accordingly,  that  integral  —  and  the  series  obtained  by  expand 
ing  it  in  powers  of  £  —  does  not  furnish  either  one  of  the  two 
values  /(£)  and  /(£)  which  the  function  f(z)  takes  on  at  these 
two  points,  but  their  sum  /(£)  -+-/(£).  To  expand  the  one  or 
the  other  of  these  two  values  in  the  neighborhood  of  the  branch 
point,  we  introduce  the  substitution  (i) ;  instead  of  integral  (6) 
we  then  investigate  the  integral 


(7) 


in  which  r  is  to  be  understood  as  that  value  of  /  which  corre 
sponds  to  z  =  £.  In  this  way  we  obtain  the  expansion  of  <f>(r) 
in  powers  of  T  with  positive  integral  exponents  ;  if  we  then 
express  T  in  terms  of  £  and  again  write  z  for  £,  we  obtain  the 
theorem : 

VI.  A  function  regular  in  the.  neighborhood  of  the  branch-point 
z  =.  o  on  the  RlEMANN'S  surf  ace  for  s  =  ~\/z,  may  be  expanded  for 
values  of  z  sufficiently  small,  in  a  convergent^  series  of  powers  of  ^/z 
with  positive  integral  exponents  —  therefore  in  powers  of  z  itself 
with  positive  exponents  which  are  integral  multiples  of  1/2. 

Since  this  series  is  obtained  by  substituting  ~Vz  for  /  in  the 
series  first  obtained,  it  is  evident  that  the  same  value  of  the  root 
is  to  be  used  in  all  of  its  terms ;  that  is,  we  are  to  understand 

m 

z*  to  be  the  mth  power  of  that  value  of  -\/z  which  we  have 
selected.  According  as  the  one  or  the  other  of  the  two  values 
of  V;s  is  taken,  we  obtain  then  the  corresponding  one  of  the 
two  values  of  the  function  at  the  points  which  are  situated  in 
the  two  sheets  of  the  surface,  one  vertically  over  the  other,  and 
which  belong  to  the  same  value  of  z. 

The  domain  of  convergence  of  this  series  is  always  bounded  by 


§  6i.   THEOREMS  AND  RIEMANX'S  SURFACE  FOR  v^     339 

a  circle ;  for,  a  circle  about  the  origin  in  the  /-plane  is  mapped 
by  substitution  (i)  into  a  circle  about  the  origin  in  the  s-plane. 

In  the  same  way,  LAURENT'S  theorem  (§  47)  is  applicable  to 
functions  which  are  regular  on  this  surface  in  the  neighborhood 
of  a  branch-point,  this  point  itself  excepted.  We  obtain  series 
arranged  according  to  powers  of  ~\/z  with  positive  and  nega 
tive  integral  exponents.  According  as  the  function  has  a  pole 
or  an  essential  singularity  at  a  branch-point,  its  expansion 
contains  a  finite  or  an  infinite  number  of  terms  with  negative 
exponents. 

Conclusions  entirely  analogous  to  these  are  valid  for  the 
neighborhood  of  the  branch-point  of  this  surface  lying  at  infin 
ity.  We  can  map  it  upon  the  neighborhood  of  the  origin  of  a 
simple  auxiliary  plane  by  the  substitution  : 

(8)  .-i,  *-=£*S 

and  we  then  regard  t  as  a  suitable  infinitesimal  of  the  first  order 
by  which  the  order  of  the  zero  or  the  infinity  of  other  functions 
is  measured.  In  this  way  z  itself  is  an  infinity  of  the  second 
order  at  infinity. 

And  Theorem  XIII  of  §  46  is  also  valid  for  every  curve 
which  completely  bounds  a  domain  of  this  two-sheeted  surface, 
inside  of  which  the  function  u  +  iv  is  regular  on  the  surface. 
Accordingly  the  second  proof  of  Theorem  IV  of  §  44,  which 
follows  Theorem  XIII,  §  46,  is  also  valid  for  this  surface,  and 
hence  the  theorem : 

VII.  A  function  everywhere  regular  on  this  two-sheeted  sur 
face  is  necessarily  a  constant. 

To  apply  further  the  conclusions  by  which  we  obtained  Theo 
rems  V  and  VI  from  IV  in  §  44,  we  would  first  try  to  form  func 
tions  which  are  regular  everywhere  on  the  surface,  with  the 


340  V.    MANY- VALUED   ANALYTIC   FUNCTIONS 

exception  of  a  single  pole,  for  which  the  terms  with  negative 
exponents  in  the  expansion  in  series  valid  for  its  neighborhood 
would  be  preassigned.  As  a  matter  of  fact,  this  is  possible  for 
the  surface  we  are  considering  but  is  not  possible  for  the  sur 
faces  determined  by  more  complicated  irrationalities ;  in  the 
application  here  we  therefore  introduce  another  method  which 
is  suitable  for  arbitrary  irrationalities. 
For  this  purpose  we  consider  the  sum 

(9)  /<»+/(*), 

in  wrhich  we  make  use  of  a  symbol  already  introduced  (§  60  a) 
and  where /(s),  the  function  to  be  investigated,  is  single-valued 
on  our  surface.  But  while  this  sum  is  single-valued  on  our 
RIEMANN'S  surface,  we  can  prove  that  it  must  also  be  single- 
valued  in  the  plane.  For,  if  we  allow  z  to  describe  a  closed 
path  in  the  plane,  for  which  there  is  also  a  corresponding 
closed  path  on  the  surface,  then  the  point  z  on  the  surface 
returns  to  z  and  the  point  z  to  ~z.  But  if  we  allow  the  point  z  to 
describe  a  closed  path  only  in  the  plane  and  not  on  the  surface, 
then  z  does  not  return  to  z  but  just  to  ~z.  If  we  start  on  the 
same  path  with  z,  we  must  return  to  z ;  for,  we  must  necessarily 
arrive  at  one  of  the  two  points  z  or  ~z :  we  cannot  return  to  ~z  as 
is  evident  if  we  trace  the  path  backwards ;  it  cannot,  therefore, 
lead  from  ~z  to  z  and  to  ~z  at  the  same  time. 

In  both  cases  the  sum  (9)  returns  to  its  initial  value  and  is 
therefore  a  single-valued  function  of  z.  As  a  matter  of  fact,  in 
the  expansion  valid  for  the  neighborhood  of  a  branch-point,  the 
terms  with  uneven  powers  of  /  disappear  because  these  terms 
in  the  expansion  of  f(z)  have  coefficients  exactly  opposite  to 
those  in  the  expansion  of /(i). 

If  we  suppose  further  that/(z)  is  regular  except  at  poles,  the 
same  supposition  holds  for/(s),  and  then  also  for  the  sum  (9); 


§  6i.  THEOREMS  AND  RI  EM  ANN'S  SURFACE  FOR  V~z     341 

this  sum  is  thus  a  rational  function  of  z  alone : 

(10)  /«+/©-*«• 

If  we  apply  to  the  product  j  •/(*)  the  conclusions  which  were 

here  applied  to  the  i  unction /(z),  we  obtain  a  second  equation 

(n)  s.f(z)-s*/(z)=r,(z\ 

in  which  r2(s)  is  also  a  rational  function  of  z  alone.     From  these 

two  equations  it  then  follows  that: 

(12)  /(,)»}n(s)+!&. 

We  have  thus  proved  the  theorem : 

VIII.  A  function  which  is  regular  on  our  surface  except  at 
poles  is  a  rational  function  of  z  and  s. 

To  apply  also  the  theorem  on  residues  to  regions  of  this  two- 
sheeted  surface,  which  contain  branch-points  in  their  interior, 
we  must  observe  that  f(z)  dz  is  transformed  by  the  substitution 
(i)  not  simply  into  $(t)dt  but  into  2  $(t}tdt\  and  by  means  of 
substitution  (8)  into  —  2  <£(/)/~V/.  It  therefore  follows  that : 

IX.  The  theorem  on  the  sum  of  the  residues  (///,  §  45)  is  also 
valid  for  functions  on  this  two-sheeted  surface,  if,  in  the  correspond 
ing  expansion  in  series,  we  consider  tlie  double  coefficient  of  f~'2,  and 
therefore  of  z~*,  as  the  residue  at  the  branch-point  in  the  finite  part 
of  the  plane,  and  the  double  coefficient  of  f2  with   the  opposite  sign, 
and  thus  again  of  z~l,  as  the  residue  at  the  branch-point  at  infinity. 

X.  But  no  such  modifications  appear  if  we  apply  Theorems  IV, 
V,  and  VI of  §  46  to  this  surface ;   for  the  substitution  (i)  and 

for  the  substitution  (8)  we  have  simply : 

and  we  have  already  agreed  that  the  order  of  the  function  at  a 
branch-point  is  to  be  determined  from  the  auxiliary  variable. 


342  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 


§  62.  The  Functions  v(*  —  d)/(z  —  b)  and  V(*  —  d)(z  —  b). 
We  study  next  the  function  : 

We  have  just  studied  the  function  -\/z  somewhat  in  detail  in  the 
last  paragraphs  and  the  discussion  of  this  apparently  more 
general  function  can  be  made  to  depend  upon  that  of  V#  by 
means  of  the  reversibly  unique  substitution 


z  —  b  z  —  i 

discussed  in  §§  14-16.     The  function 

(3)  *  =  V7 

determines  a  certain  surface  upon  the  /-sphere ;  by  means  of 
the  substitution  (2)  we  now  transform  this  /-sphere  together 
with  the  surface  spread  out  over  it  into  the  z-sphere  and  the 
corresponding  two-sheeted  surface  covering  it ;  this  latter  sur 
face  is  now  sufficient  to  represent  geometrically  function  (i) 
and  its  branches,  since  this  function  is  a  single-valued  and  con 
tinuous  function  of  position  on  this  surface.  The  branch-points 
z'  =  o  and  z'  =  oo  of  the  first  surface  correspond  to  the  branch 
points  z  =  a  and  z  —  b  of  the  latter  surface  ;  the  half-axis  of 
negative  real  numbers  according  to  IV,  §  14,  corresponds  to  an 
arc  of  a  circle  connecting  these  two  points  and  passing  also 
through  the  point  z  =  ^(a  +  b)  (corresponding  to  z'  =  —  i)  ;  that 
is,  to  the  straight  line  ab. 
The  function : 

is  also  single-valued  on  the  surface  thus  constructed ;  this  is 


§62.    FUNCTIONS    V(«-  a)/(z-  J)    AND    V(z-0)(z-£)       343 

evident  when  it  is  put  in  the  form  : 

(5)  *  =  («-*)*». 

This  form  shows  that  o-  is  a  rational  function  of  z  and  s  ;  and 
conversely  that  : 


is  a  rational  function  of  2  and  o-.     This  is  equivalent  to  saying 
that  : 


I.  V(z  —  a]j(z  —  b]  and  V(z  —  #)(z  —  b}  are  irrationalities  of 
the  same  class. 

We  can,  of  course,  construct  the  RIEMANN'S  surface  for  a- 
directly  without  making  use  of  s.  For  this  purpose  we  start 
from  the  fact  that  the  equation 


(7) 

is  a  complete  equation  between  many-valued  functions,  in  the  sense 
explained  in  VII,  §  56  that  every  value  of  the  right-hand  side 
is  equal  to  a  value  of  the  left-hand  side  and  conversely  ;  it  fol 
lows  from  this  simply  that,  for  given  values  of  z1  and  %  each 
side  has  two  and  only  two  values  (not  the  possibility  that  the 
right  side  has  four  values).  Consequently  the  change  of  value 
of  V(2  —  a)(z  —  I)}  as  z  varies  continuously  is  made  clearer  by 
observing  the  change  in  value  of  each  of  the  two  factors  ^/z  —  a 
and  ^/z  —  b.  But  that  is  simply  the  question  treated  in  §§  58- 
6  1  only  that  the  point  a  (and  b)  now  appears  in  place  of  the 
origin  :  V  'z  —  a  changes  its  sign  if  z  makes  a  circuit  about  the 

*  The  reader  is  already  familiar  with  this  idea  in  connection  with  integration,  for 

it  is  used  to  reduce    i  ^/(z  —  a)  •  (z  —  d)  dz  to  the  integral  of  a  rational  function. 

_  /*  r2 

The  transformation   V(z  —  a)  •  (z  —  b)  =  (z  —  b}s   leads  to    \  (a—t>)*     -  -  —  ds. 

-S.E.R. 


344  v-    MANY-VALUED   ANALYTIC   FUNCTIONS 

point  a,  ~\/z  —  b  so  changes  if  z  encircles  the  point  b.  The  prod 
uct  then  changes  its  sign  or  remains  unchanged  according  as 
the  path  of  z  makes  a  total*  of  an  odd  or  an  even  number  of 
circuits  about  the  points  a  and  b.  If  this  number  is  uneven,  we 
could  stop  the  corresponding  paths  by  connecting  a  and  b  by  a 
line  and  not  allowing  z  to  cross  this  line  ;  for  then  it  can  describe 
only  such  paths  which  encircle  one  of  these  two  points  as  often 
as  the  other.  If  therefore  we  make  an  incision  in  the  plane 
along  this  line,  a  branch  of  the  function  is  denned  to  be  single- 
valued  on  the  plane  cut  in  this  way ;  let  us  now  take  two  planes 
(or  spheres),  each  of  which  has  been  cut  in  this  manner,  and 
fasten  them  together  crosswise  along  this  cut.  We  thus  obtain 
a  RIEMANN'S  surface,  on  which  the  function  under  consideration 
is  a  single-valued  and  continuous  function  of  position  —  exactly 
the  same  surface  which  was  obtained  above  in  other  ways. 

(It  remains  to  be  mentioned  that  the  point  z  —  oc  is  not  a 
branch-point  of  the  surface  ;  to  be  sure,  if  we  encircle  it,  each  of 
the  factors  changes  its  sign  and  hence  the  function  itself  does 
not  change  its  sign.  The  expansion  of  the  function  for  the 
neighborhood  of  the  point  z  =  oo  is,  in  one  sheet, 

/fiv  a  +  b      gL-iab  +  P  , 

(8)  ar=Z —  -  ~^~  "5 

in  the  other  sheet, 

,  x  ,  a  +  b  ,  a2  —  2  ab  +  ^ 

(9)  a  =  -Z  +  -     -  +  -  . 


§  62  a.     Rational  Functions  of  z  and  o-  =  V(*  —  a)(z  —  b). 
By  solving  equation  (i)  of  §  62  for  z  we  obtain 


*  Italics  by  the  translator, 


§  62a.    RATIONAL  FUNCTIONS  OF  z  AND  <r  =  V(z-a)(z-6)       345 

Thus  s  isa.  rational  function  of  s.  and  accordingly  even*  rational 
function  of  2  and  s  may  be  expressed  here,  as  in  §  60  a,  as  a 
rational  function  of  s  alone.  It  follows  from  equation  (5),  §  62, 
that  ever}*  rational  function  of  cr  and  2  can  be  represented  as  a 
rational  function  of  a  single  variable  s,  or,  as  we  are  accustomed 
to  saying,  can  be  4<  rationalized  by  introducing  s.  "  But  we  will 
investigate  these  rational  functions  directly  without  making  use 
of  this  method  of  reduction  since  it  cannot  be  applied  to  com 
plicated  algebraic  functions. 

We  can  now  put.  as  in  §  60  a,  even*  rational  function  of  z  and 
a-  in  the  form  : 


in  which  gv  glf  g.,  signify  rational  integral  functions  of  z  alone  ; 
we  suppose  also,  that  g0,  g^  g.2  have  no  common  divisor. 

We  wish  to  investigate  how  such  a  function  behaves  in  the 
neighborhood  of  any  point  on  the  RIEMAXX'S  surface  of  <r. 
First,  let  z  =•  z0  be  an  ordinary  point  on  the  surface  ;  that  is,  one 
lying  at  a  finite  distance  from  the  origin  and  not  a  branch-point, 
and  <TO  the  corresponding  value  of  a-.  We  can  then  expand  <r 
by  the  binomial  theorem  in  the  following  series  of  powers  of 

(3)  *=»-* 

convergent  in  a  sufficiently  small  neighborhood  of  z0: 

(4)  <r  =  « 


—a      z  — 


If  we  expand  g<£z),  g\(z),  g»(z)  in  the  same  way  in  powers  of  /, 
replace  them  in  the  expression  and  rearrange,  R  is  represented 
as  follows  as  the  quotient  of  two  power  series  : 

/x  p  _. 


346  V.    MANY-VALUED   ANALYTIC  FUNCTIONS 

The  following  cases  present  themselves  : 

1 .  If  ft  =f=.  o,  J?  is  a  function  of  /,  and  therefore  of  z,  regular 
in  the  neighborhood  of  /=o. 

2.  If  ft  =  o,  but  «0  ^  o,  let  ft  be  the  first  coefficient  different 
from  zero  in  the  denominator.     Then  R(z,  <r)  is  equal   to  t~k 
times  a  function  which  is  regular  at  /=  o  ;  we  say  therefore  that 
in  this  case  the  function  R  has  a  pole  of  the  £th  order  at  z  =  ZQ  , 
a-  =  O-Q. 

3.  If  «0  and  ft  are  both  equal  to  zero,  R  is  indeterminate  at 
f=o.     But  we  can  remove  this  ambiguity  as  in  §  20  by  dividing 
numerator  and  denominator  by  a  suitable  power  of  /;  in   this 
way  this  case  reduces  to  one  of  the   cases   (i)  or  (2)  already 
discussed. 

Second,  let  the  point  be  a  branch-point  and  we  investigate  the 
behavior  of  R  in  the  neighborhood  of  such  a  point,  say  z  =  a. 
Put 
(6)  z—a=t2,     -Vz  —  a  =  /; 

in  this  way  the  neighborhood  of  the  branch-point  in  both  sheets 
of  the  surface  is  mapped  upon  the  simple  neighborhood  of  the 
origin  of  the  /-plane.  The  representation  is  determined  when 
ever  the  sign  of  the  root  is  fixed  in  (6)  for  one  value  of  z  in  the 
given  neighborhood.  For  two  values  of  /  which  are  equal  but 
opposite  in  sign,  there  are  points  of  the  surface  which  lie  exactly 
vertical  to  each  other  in  the  two  shepts. 
Then 

(7) 


and  R  is  thus  represented  for  this  case  also  as  the  quotient  of 
two  power  series  in  powers  of  this  auxiliary  variable  /  with  in 
tegral  exponents.  In  this  connection  corresponding  conclusions 
can  then  be  made ;  and,  too,  it  is  suitable,  as  in  §  6 1,  to  deter- 


§62  a.    RATIONAL  FUNCTIONS  OF  z  AND  <r  =  V(*  -  a)(z  -  b}    347 

mine  whether  a  function  R(z,  or)  is  regular  at  a  branch-point 
when  it  is  a  regular  function  of  /  at  such  a  point  and,  in  other 
cases,  to  determine  the  order  of  the  infinity  from  the  exponent 
of  /  (not  from  the  exponent  of  z  —  a  itself). 

Third.  In  order  to  investigate  the  behavior  of  R  (z,  <r)  for  in 
definitely  large  values  of  z,  let  us  put 

(8)  z  =  r\ 

This  gives  the  two  expansions  (8)  and  (9)  of  §  62,  corresponding 
to  the  two  points  of  the  RIEMANN'S  surface  at  infinity ;  there  is 
an  expansion  of  R  in  powers  of  /  for  each  of  these  points,  and  the 
order  of  the  infinity  is  again  determined  from  the  lowest  expo 
nent  of  /  appearing  in  the  expansion.  Thus,  for  example,  z  and 
o-  become  infinite  of  the  first  order  in  both  sheets  of  the  surface. 
The  result  of  the  investigation  is  therefore  that : 

I.  A  rational  function  of  z  and  a-  is  regular  over  the  entire 
RIEMANN'S  surface  of  a-  except  at  poles. 

(That  there  can  be  only  a  finite  number  of  poles,  follows  from 
the  fact  that  they  can  only,  but  not  necessarily  must,  appear 
where  g$(z)  =  o.) 

The  converse  of  Theorem  I  is  proved  as  in  §  61. 

There  is  a  certain  interest  in  the  question  whether  there  are 
rational  functions  of  z  and  a-  which  become  infinite  at  only  one 
point  of  the  surface,  and  of  the  first  order  at  this  point;  and  fur 
ther  whether  this  point  can  be  chosen  arbitrarily.  This  question 
is  answered  at  once  by  "  rationalizing  "  the  function,  but  we 
shall  attack  the  problem  directly.  In  order  to  simplify  the  pro 
cess  let  us  put  a  =  i ,  b  =  —  i ;  we  can  at  once  reduce  the  more 
general  case  to  this  one  by  means  of  a  reversibly  unique  trans 
formation  of  the  2-plane  according  to  II,  §  15. 

If  we  wish  to  find  a  rational  function  of  z  and  o-  =  Vz2  —  i 
which  has  an  infinity  at  only  one  finite  point  z  =  a  distinct 


348  V.    MANY-VALUED   ANALYTIC  FUNCTIONS 

from  the  branch-points,  and  in  fact,  only  in  one  sheet  and  only 
of  the  first  order,  it  follows  that  the  denominator^^)  in  (2)  can 
have  no  factors  of  the  first  degree  other  than  z—  a.  For,  if  the 
denominator  were  divisible  by  z  — 13,  then  the  function  J?  for 
z  —  (3  would  become  infinite  at  one  or  the  other  of  these  points  of 
the  surface,  provided  that  the  numerator  would  not  also  become 
zero  for  both  values  of  a.  But  since  cr  does  not  by  hypothesis 
become  zero  for  z  =  /?,  the  numerator  is  zero  only  when  g^z) 
and  gi(z)  are  both  zero  for  z=/3,  that  is,  when  both  are  divisible 
by  z  —  ft.  But  then  gQ ,  gl ,  g.2  would  all  three  have  the  same 
common  divisor  z  —  (3  contrary  to  hypothesis. 

In  the  same  way  it  can  be  shown  that  g^(z)  is  not  divisible  by 
powers  of  (z  —  a)  higher  than  the  first,  when  the  function  jR  for 
z  =  a  does  not  become  infinite  of  higher  order  than  the  first  in 
either  one  of  the  two  sheets  of  the  surface. 

We  can,  therefore,  take  g«(z)  =  z  —  a,  since  a  constant  factor 
can  be  divided  out  in  the  numerator. 

For  z  =  a  there  are  two  values  of  cr ;  if  we  call  <ra  a  certain 
one  of  them,  —  o-a  will  be  the  other  one.  And  if  R  becomes 
infinite  for  (a,  cra),  but  not  for  (a,  —  o-a),  then  the  numerator  of 
(2)  must  be  zero  for  (a,  —  <ra),  and  thus 

(9)  #>(«)  -  <ra  '  £i(«)  =  o. 

This  is  a  linear  homogeneous  equation  between  the  coefficients 
of  go  and  glt  If  it  is  satisfied,  JR  will  not  become  infinite  at  the 
point  (z  =  a,  <r  =  —  <ra)  as  is  shown  by  a  procedure  similar  to 
the  one  at  the  beginning  of  this  section. 

Finally,  we  must  also  be  certain  that  our  function  remains 
finite  at  infinity  in  both  sheets.  The  denominator  has  an 
infinity  there  of  the  first  order  ;  and  hence  we  must  be  careful 
that  the  numerator  does  not  become  infinite  of  higher  order. 
Accordingly,  g0  -f-  agl  and  g0  —  agl  must  not  become  infinite  of 


§  62  a.    RATIONAL  FUNCTIONS  OF  z  AND  cr  =  \/(z  -  a)  (2  -  6)      349 

higher  order  than  the  first,  where  o-  represents  one  and  —  a- 
represents  the  other  of  the  two  expansions  (8),  (9),  §  62.  Addi 
tion  and  subtraction  shows  that  gQ  and  o^  must  not  become 
infinite  of  higher  order  than  the  first  and  therefore  gl  in  general 
must  not.  That  is,  gQ  must  be  a  linear  function  of  z,  and  g±  a 
constant  ;  accordingly  we  put 

(10)  g0  =  (Az  +  £)<ra,   g,=  C. 

Between  these  three  constants  the  relation 
(n)  Aa  +  £—C  =  o 

must  exist  on  account  of  (9)  ;  one  of  these  constants  is  expres 
sible  in  terms  of  the  other  two  by  (n).  In  this  way  we  obtain 
the  result  : 

II.  Every  rational  function  of  z  and  a  which  becomes  infinite  at 
only  the  one  point  (z  =  «,  o-  =  <ra)  of  the  surface,  and  only  of  the 
first  order  at  this  point,  has  the  form 

(Az  +  ^)tra  +  (Aa  +  B)<r_ 


z  —  a 

Conversely,  every  function  of  this  form  has  the  required  property, 
omitting,  of  course,  the  trivial  case  Aa  +  B  =  o,  in  which  it 
reduces  to  a  constant. 

Further,  if  we  wish  to  form  a  function  which  becomes  infinite 
only  at  the  branch-point  z  =  —  i,  and  only  of  the  first  order 
there,  we  see  as  in  the  previous  case  that  £0(2)  must  equal  z-\-i. 
But  this  function  becomes  zero  of  the  second  order  at  the 
branch-point  ;  we  must  therefore  make  provision  that  the 
numerator  becomes  zero  and  of  the  first  order  there.  This 
requires  that  gQ(z)  be  divisible  by  z  -f-  i  ;  and  if  we  consider  as 
before  the  behavior  of  the  function  at  infinity,  we  find  : 


350  V.    MANY-  VALUED   ANALYTIC   FUNCTIONS 

III.  Every  rational  function  of  z  and  a-,  which  becomes  infinite 
only  at  the  branch-point  z  =  —  I  and  there  only  of  the  first  order, 
has  the  form  : 

(13)  A(z+  T)  +  C(T. 

z  H-  i 

or,  by  using  ^Jz  -f-  /,  it  takes  the  form 


(14)  A  +  C-  V(*  -i)/(*  +  i). 

Finally,  to  form  a  function  which  becomes  infinite  only  at 
infinity  but  there  only  in  one  sheet  and  only  of  the  first  order, 
corresponding  conclusions  as  in  the  first  two  cases  show  that 
gz(z)  reduces  to  a  constant,  that  gQ(z)  is  a  linear  function  and 
that  gtf  must  also  be  a  constant.  If,  therefore,  R  is  to  become 
infinite  when  we  use  the  expansion  (8),  §  62  for  <r,  but  not  when 
(9),  §  62  is  so  used,  a  linear  equation  between  the  constants 
exists. 

Hence  the  theorem  : 

IV.  Every  rational  function  of  z  and  <r,  which  becomes  infinite 
only  at  infinity  and  there  only  in  one  sheet  and  only  of  the  first 
order,  has  the  form  : 

(15)  A(z  +  <r)+B. 

Moreover,  the  constants  at  our  disposal  in  (12),  (14),  (15)  can 
be  so  chosen  that  the  function  under  consideration  becomes 
zero  at  a  preassigned  point.  Particular  interest  attaches  to  the 
function 


(16)  *=V(* -i)/(*+ i), 

which  becomes  zero  at  one  branch-point   and   infinite   at  the 
other  ;  to  the  function 

(17)  u  =  z  +  <r, 


§62b.    INTEGRALS   OF   RATIONAL   FUNCTIONS   OF  z         351 

which  at  infinity  becomes  zero  in  one  sheet  and  infinite  in  the 
other  ;  and  then  also  to  the  function 

(.8)  .          *  =  1-L> 


which  at  z  =  o  becomes  zero  in  one  sheet  and  infinite  in  the 
other.  The  first  is  precisely  the  function  designated  by  s  in  §^62. 
According  to  Theorem  VI,  §  46,  which  also  holds  here  (cf.  X, 
§  61),  each  of  the  functions  considered  has  the  property  that  it 
takes  on  in  general  each  value  on  the  surface  once  and  only 
once.  It  follows  from  this  that  each  of  them  is  a  single-valued 
analytic  function  of  each  of  the  others,  which  takes  on  each 
value  once  and  only  once.  And  from  this  it  follows  further  that 
each  of  these  functions  is  a  linear  fractional  function  of  each  of 
the  others.  For  example  : 

,    v  i  +  «f  it—  i      ,       i  —  is      .     i  —  in 

(19)  u  =  -    -,    f  =  -    — ,    A=-  — -  =  *.-— -. 

I  —  S  U  -\-  I  I  +  M  I  -f  III 

And  it  follows  further  that  any  function  of  the  surface  is  a  single- 
valued  function  of  each  of  these  auxiliary  variables.  We  have 
thus  returned  to  that  starting  point  which  was  intentionally 
avoided  at  the  outset. 

§  62  b.     Integrals  of  Rational  Functions  of  z,  and  the  Square  Root 
of  a  Rational  Integral  Function  of  z  of  the  Second  Degree 

Since  all  rational  functions  of  z  and  or  used  above  may  be 
represented  as  rational  functions  of  an  auxiliary  variable,  it  fol 
lows  that  every  integral  of  such  a  function  can  be  transformed 
into  an  integral  of  a  rational  function  and  therefore  can  be  ex 
pressed  in  terms  of  rational  functions  and  the  logarithms  of  such 
functions.  In  this  any  of  the  functions,  which  were  considered 
in  the  latter  part  of  the  previous  paragraph  and  which  take  on 


352  V.    MANY- VALUED   ANALYTIC   FUNCTIONS 

any  value  once  and  only  once  on  the  surface,  can  be  used  as 
auxiliary  variable.  In  the  elements  of  the  integral  calculus  we 
prefer  to  use  the  three  functions  (16),  (17),  (18)  of  §  62  a,  since 
they  enable  us  to  perform  the  processes  most  conveniently. 

But  we  will  also  consider  the  application  of  these  integrals  of 
rational  functions  of  z  and  or  directly  to  the  surface.  It  follows 
from  IX,  §  61,  that: 

I.  Such  an  integral  is  a  single-valued  function  of  its  upper  limit, 
provided  that  the  path  of  integration  lies  entirely  in  a  simply  con 
nected  part  of  the  surface  which  contains  no  point  at  which  the 
residue  of  the  function  is  different from  zero. 

But  if  the  path  of  integration  taken  in  the  positive  sense  en 
circles  such  a  point,  a  new  value  of  the  integral  is  obtained 
which  is  greater  than  the  former  value  by  2  ?r/  times  the  corre 
sponding  residue.  Hence : 

II.  If  the  path  of  integration  for  such  an   integral  is  entirely 
arbitrary,  we  obtain  an  infinitely  many-valued  function  ;  all  of  its 
values  follow  from  one  of  them  by  the  addition  of  integral  multiples 
of  a    certain    "  modulus  of  periodicity"     Each    such    modulus  of 
periodicity  is  equal  to  2  iri  times  the  residue  of  the  function  to  be 
integrated. 

We  now  classify  the  integrals  according  to  the  kind  and  num 
ber  of  their  points  of  discontinuity.  In  this  connection  we 
notice  that :  at  each  finite  point  at  which  the  function  to  be 
integrated  is  finite,  the  integral  is  also  finite  ;  at  each  finite 
point  which  is  not  a  branch-point  and  at  which  the  function 
becomes  infinite,  the  integral  also  becomes  infinite ;  but  at  a 
branch-point  the  integral  can  remain  finite  even  if  the  function 
to  be  integrated  becomes  infinite.  For,  from  the  substitution  (6), 
§  62  a,  we  obtain 
(i)  dz  =  2tdt. 


§  62  b.   SQUARE  ROOT  OF  A  2cl  DEGREE  FUNCTION       353 

If  we  then  use  /  as  the  variable  of  integration,  an  additional 
factor  /  is  obtained  under  the  sign  of  integration,  and  the  inte 
gral  remains  finite  provided  that  the  infinity  of  the  function 
to  be  integrated  is  not  of  order  higher  than  the  first.  Corre 
sponding  considerations  show  that  at  an  infinitely  distant  point, 
the  integral  remains  finite  when  the  function  to  be  integrated 
becomes  zero  of  order  higher  than  the  first. 

We  ask  next  whether  there  are  integrals  of  rational  functions 
of  z  and  <r  which  are  nowhere  infinite.  (Theorem  IV,  §  44  would 
not  contradict  this  statement,  since  it  only  treats  of  single-valued 

analytic  functions  of  z.)  If  (  R(z,  <r)dz  remains  finite  every 
where,  R(z,  a-)  must 

i st.  Be  everywhere  finite,  except  at  the  branch-points  where 
it  might  become  infinite  of  the  first  order  ; 

2d.  Become  zero  of  higher  order  than  the  first  at  infinity  in 
both  sheets. 

The  product 

(2)  *  •  £(*,  <r) 

must  then  be  finite  everywhere  and  zero  at  infinity.  But  from 
§  62  a  it  follows  that  there  is  no  rational  function  of  z  and  <r 
which  is  finite  everywhere.  It  then  follows  that : 

III.  There  is  no  integral  which  is  finite  everywhere  on  the 
R  IE  MANN'S  surface  of  v  =  Vz2  —  I. 

We  discuss  next  integrals  which  have  only  logarithmic  dis 
continuities.  Since  the  sum  of  the  residues  on  the  whole  sur 
face  must  be  equal  to  zero  (the  proof  of  Theorem  VI,  §  45,  is 
applicable  here  without  change),  such  an  integral  must  have  at 
least  two  points  of  discontinuity ;  we  wish  to  form  an  integral 
having  such  discontinuities  at  only  two  ordinary  points  (z^  o^) 

and  (zz,  <r2)  on  the  surface.  If  I  R(z,  o-)rfz  is  such  an  integral, 
the  function  R  must  have  the  following  properties : 


354  v-    MANY-VALUED   ANALYTIC   FUNCTIONS 

It  must  be  finite  everywhere,  except  at  the  two  given  points  and 
at  the  branch-points,  where  it  may  be  infinite  of  the  first  order  ; 

It  must  be  zero  of  an  order  higher  than  the  first  at  infinity  in 
both  sheets. 

The  product  aR  must  therefore  have  the  following  properties  : 

It  must  be  finite  everywhere  on  the  finite  part  of  the  surface 
except  at  the  two  points  (zlt  o^)  and  (z2,  o-2)  where  it  may  be  in 
finite  of  the  first  order  ; 

It  must  be  zero  at  infinity. 

We  can  represent  such  a  function  as  the  sum  of  two  functions, 
each  of  which  becomes  infinite  at  one  of  the  given  points  and 
both  become  zero  at  infinity  in  the  same  sheet  ;  therefore,  ac 
cording  to  (12),  §  62  a,  the  function  takes  the  form  : 

(3)  ^{i 


I  Z  —  z\   }  (  Z  —  Z2  ) 

and  the  constants  A,  B  can  be  so  determined  that  the  sum  be 
comes  zero  at  infinity  in  the  other  sheet  also.     We  thus  obtain 


Z  —  %         Z  — 

as  the  desired  form  of  an  integral  having  only  two  logarithmic 
discontinuities. 

The  constant  A  can  also  be  so  determined  that  the  residue 
at  one  point  of  discontinuity  is  equal  to  +  i,  at  the  other  equal 
to  —  i  ;  for  this  purpose  we  must  take  A  =  1/2. 

If  we  then  introduce  as  the  variable  of  integration  the  func 
tion  designated  by  u  in  (17),  §  62  a,  and  call  u^  and  u2  the 
values  which  this  function  takes  on  at  the  two  points  (zlf  ox)  and 
fe>  °"2)>  we  obtain  : 


u  —  u       u  —  u2  u  — 


§  62  c.    THE   FUNCTION   z  =  w  +  zVi  -  w*  355 

If  \\e  assume  the  two  points  (%,  o^)  and  (s2,  0*2)  to  be  coinci 
dent  and  then  divide  by  u^—  u2,  we  can  obtain  from  (5)  the  fol 
lowing  integral  which  becomes  infinite  at  only  one  place  but 
algebraically  of  the  first  order  at  this  place  : 

du  i  ^, 


s 


Repetition  of  this  process  leads  then  to  integrals  which  be 
come  infinite  of  the  second,  third,  etc.,  order  at  a  preassigned 
place  and  which  have  coefficients  preassigned. 

In  this  process  it  is  assumed  that  the  singular  points  are  dif 
ferent  from  the  branch-points  and  lie  on  the  finite  part  of  the 
surface ;  in  fact,  we  would  encounter  no  fundamental  difficulties 
in  a  corresponding  treatment  of  the  cases  thus  excluded.  But 
it  is  unnecessary  to  enter  into  a  discussion  of  this  point  since 
the  whole  investigation  can  be  arranged  here  (except  for  more 
complicated  irrationalities)  to  depend  upon  rational  functions  at 
the  beginning  by  introducing  //  as  independent  variable. 

The  most  general  integral  of  a  rational  function  of  z  and  <r 
can  then  be  represented  as  a  sum  of  integrals  of  the  special 
form  considered,  with  suitable  numerical  coefficients.  This  fol 
lows  from  the  fact  that  the  difference  of  two  integrals,  which 
become  infinite  in  the  same  manner,  is  an  integral  which  never 
becomes  infinite  and  is  therefore  a  constant  according  to  III. 

§  62  c.    The  Function  z  =  w  +  /  V*  -  w* 

According  to  the  definition  of  the  square  root  of  a  complex 
number,  the  solution  of  a  quadratic  equation  with  complex  coeffi 
cients  is  found  just  as  we  obtained  the  solution  of  the  quadratic 
equation  with  real  coefficients  in  elementary  algebra.  Thus, 
for  example,  if  we  solve  for  z  the  equation  discussed  in  §  21  a, 


356  V.    MANY- VALUED   ANALYTIC   FUNCTIONS 

or 

(2)  Z2  —  2  ZW  +  I  =  O, 

we  obtain : 

(3)  z  =  w -{-  Vw2  —  i. 

This  function  is  complex  for  real  values  of  w  whose  absolute 
value  is  less  than  i  ;  this  is  more  evident  by  writing 


(4)  0  =  w  +  iVi  —  w*-\ 

but  we  must  keep  in  mind  that  the  principal  value  of  the  square 
root  in  (3)  does  not  also  furnish  the  principal  value  of  the  square 
root  in  (4)  for  all  values  of  w. 

As  a  special  case  of  the  results  of  §  62,  it  follows  that  the 
RIEMANN'S  surface  extended  over  the  w-plane  and  determined 
by  this  function,  consists  of  two  sheets  which  are  united  at  the 
two  branch-points  a/ =  +  i  and  «/  =  — i.  We  connect  these 
two  branch-points  by  a  cut ;  this  is,  perhaps,  most  conveniently 
done  by  drawing  the  cut  from  both  of  these  points  along  the 
w-axis  of  real  numbers  to  infinity.  The  origin  is,  therefore,  not 
on  the  cut ;  consequently  we  distinguish  between  the  two  sheets 
of  the  surface  by  determining  what  value  w  shall  take  on  at  the 
origin  in  each  of  the  two  sheets.  Thus  we  name  arbitrarily  the 
first  sheet  that  one  for  which  z  =  i  and  w  =  o,  the  second  sheet 
that  one  for  which  z  =  —  i  and  w  =  o.  The  values  of  z  at  the 
remaining  points  of  both  sheets  are  obtained  by  proceeding 
continuously  ;  that  is  to  say,  in  the  neighborhood  of  the  origin 
the  principal  value  of  the  square  root  in  (4)  will  be  taken  in 
the  first  sheet,  but  the  opposite  value  in  the  second  sheet.  But 
we  can  by  no  means  yet  conclude  that  this  is  true  throughout 
the  whole  extent  of  both  sheets. 

The  investigation  of  this  function  is  very  much  simplified 
from  the  fact  that  its  inverse  is  a  single-valued  function  of  z 
and  that  we  have  already  investigated  this  inverse  function  in 


§  62  c.    THE   FUNCTION  z  =  w  +  zVi  - 


357 


detail  in  §  2 1  a.  We  need  only  to  make  the  results  thus  known 
still  clearer  by  employing  the  RIEMANN'S  surface  introduced  in 
the  meantime. 

We  notice  next  that  real  values  of  z  correspond  to  the  points 
of  the  branch-cut ;  and,  in  fact,  to  each  such  value  of  w  corre 
spond  two  values  of  2,  one  of  which  lies  inside  of  the  unit  circle 
and  the  other  outside  of  it,  since  the  product  of  the  roots  of 
equation  (2)  is  equal  to  i.  Only  one  value  of  z  corresponds  to 
each  of  the  branch-points ;  to  w  =  -j-  i  corresponds  z  =  -f-  i  and 
for  w  =  —  i,  5  =  —  i.  Conversely,  one  point  of  the  branch-cut 
corresponds  to  each  real  value  of  z.  Therefore,  one  sheet  of 
the  surface  corresponds  to  the  positive,  the  other  sheet  to  the 
negative,  s-half-plane.  But  which  sheet  corresponds  to  which 
half-plane  is  not  now  an  arbitrary  arrangement,  since  we  have 
already  disposed  of  this  question  by  giving  to  z  the  value  -f  / 
at  the  origin  in  the  first  sheet ;  in  this  wray  the  first  sheet  must 
correspond  to  the  positive  half-plane,  and,  therefore,  the  other 
sheet  to  the  negative  half-plane.  And  thus,  too,  it  is  determined 
how  the  two  letters  which  are  assigned  to  one  region  in  Figure 


J^ 

4                Q 

c 

^           —  ~ 

St                                jL/{ 

"T*" 

FIG 

35  a                          FIG 

•35* 

First  sheet  over  the          Second  sheet  over 

z0-plane.                        the  t£»-plane. 

M 


C' 


A 


FIG.  13,?, 


13  h  are  arranged  on  the  two  sheets.  Figure  13  e  is  repeated 
here  in  connection  with  the  two  other  figures  for  the  purpose 
of  better  comparison. 


358  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

The  relations  in  the  neighborhood  of  the  point  at  infinity  in 
both  sheets  of  the  w-plane  are  of  interest  ;  developing  the  radi 
cal  in  decreasing  powers  of  w,  we  obtain  : 

(5)      vW2  —  i  =  w  Vi  —  i/^2  =  ±  w  \  i  --  :  -  +  —  —  —  •••  I  ; 

[         2W1      8w4  J 

the  one  value  of  z  is  therefore  equal  to 


and  the  other  value  of  z  is  equal  to 

(7)  2W-—  +^-3-  +  ----  . 

2  W        8w3 

The  first  one  of  these  values  becomes  zero  at  infinity  and  the 
other  one  is  infinite  there  ;  as  the  figures  show,  the  first  de 
velopment  holds  for  that  "  part  of  the  surface  "  (cf.  end  of  §  59) 
which  consists  of  the  lower  half-plane  of  the  first  sheet  and  the 
upper  half-plane  of  the  second,  while  the  other  development 
holds  for  the  remaining  part  of  the  surface. 

If  we  had  made  the  branch-cut  along  the  shortest  line  con 
necting  the  branch-points  instead  of  along  the  two  segments  of 
the  real  w-axis  external  to  these  points,  the  regions  Alt  A%,  Z>u 
Z>2,  would  have  represented  the  one  sheet  of  the  surface,  the 
regions  B^,  _Z?2>  6\  ,  C2,  the  other  sheet;  and  therefore  the  one 
sheet  would  have  corresponded  to  the  inside  of  the  unit  circle 
of  the  2-plane,  the  other  sheet  to  the  outside  of  this  circle. 

To  go  further  into  details  we  would  introduce  in  the  figures 
the  circles  and  straight  lines,  the  confocal  ellipses  and  hyper 
bolas  which  were  used  for  a  similar  purpose  in  §  21  a. 

According  to  §  62,  z  may  be  rationalized  by  the  substitution 


§  62  c.    THE  FUNCTION  z  =  w  +  iVi  -  w2  359 

we  find  : 


i  — 


This  is  a  fractional  function  of  the  first  degree  ;  conversely,  s  is 
also  expressible  rationally  in  terms  of  z  : 

i  —  z 

(IO)  '  =  7TV 

and  s  could  be  chosen  instead  of  z  as  that  function  of  the  surface 
by  which  all  other  functions  of  the  surface  are  expressed  ration 
ally.  We  find,  as  a  matter  of  fact,  that 


2  Z 


It  is  now  easy  to  answer  the  question  heretofore  postponed 
concerning  the  region  of  this  surface  to  which  the  principal  value 
of  the  square  root  belongs.  This  region  must  be  bounded  by 
the  line  or  lines  along  which  the  square  root  is  purely  imaginary. 
This  is  true  along  the  s-axis  of  reals  and  along  no  other  lines  ; 
the  branch-cut  in  the  ay-plane  corresponds  to  it.  Therefore  the 
principal  value  is  attached  to  the  entire  first  sheet. 

Since  we  have  already  discussed  the  exponential  and  the 
trigonometric  functions  of  complex  argument,  the  relation  be 
tween  z  and  w  can  now  be  made  clear  by  the  introduction  of 
other  auxiliary  variables  than  the  Z  and  W  which  were  used 
in  §  2  1  a.  Thus  if  we  put 


it  follows  from  (4)  and  (16),  §  40,  that 

(13)  z  =  e\ 

In  fact,  by  means  of  these  equations  the  concentric  circles 
about  the  origin  and  the  rays  through  the  origin  in  the  z-plane 


360  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

correspond  respectively,  according  to  III,  §  42,  to  the  parallels 
to  the  axes  in  the  ^-plane,  and,  according  to  IV,  §  42,  these 
parallels  correspond  to  the  confocal  ellipses  and  hyperbolas  in 
the  ze/-plane. 

§  62  d.     The  Function  Sin-1  w 

We  wish  now  to  define  the  function  s'm^1  w  just  as  for  real 
variables  by  the  integral 

(i)  J  =  sin'1  w  =    I     — — — — ; 

Jo       y'j    _  W2 

to  do  this  two  additional  specifications  are  necessary ;  we  must 
make  provision  concerning  the  path  of  integration  to  be  chosen 
and  concerning  the  value  to  be  given  to  the  square  root. 

We  specify  the  path  of  integration  to  be  entirely  arbitrary, 
except  that  it  must  not  go  through  one  of  the  branch-points  since 
doing  so  would  lead  into  difficulties.  However,  we  need  not  ex 
clude  the  case  where  the  upper  limit  takes  on  one  of  these 
values,  since  it  can  be  shown  just  as  for  real  variables  that  the 
integral  approaches  in  this  case  a  definite  finite  limit.  We  must 
not  suppose  however  that  the  value  of  the  integral  is  entirely  in 
dependent  of  the  path  of  integration ;  the  symbol  s'm~lw  is  de 
fined  by  equation  (i)  to  be  many-valued,  but  we  will  consider  all 
the  values  which  it  takes  on  according  to  the  definition  as 
belonging  to  one  and  the  same  many-valued  function  of  w. 

The  sign  to  be  given  the  root  can  be  fixed  arbitrarily ;  and 
we  agree  that  the  value  4-  i  shall  be  attached  to  it  at  the  lower 
limit  of  integration.  But  in  so  doing  its  value  is  fixed  for  the 
whole  of  the  remainder  of  the  path  of  integration,  provided  the 
values  change  continuously  along  the  path.  Any  doubt  con 
cerning  the  value  of  the  root  could  only  arise  when  the  path  of 
integration  goes  through  one  of  the  points  (0  =  4-1  or  w  =  —  i  ; 
but  this  possibility  is  already  excluded.  We  determine  most 


§62d.    THE   FUNCTION   Sin-1™  361 

simply  what  value  of  the  square  root  obtains  at  any  point  of  the 
path  of  integration,  if  we  make  use  of  the  RIEM ANN'S  surface 
already  introduced  in  the  previous  paragraphs  upon  which  this 
square  root  is  a  single-valued  and  in  general  continuous  func 
tion  of  position,  and  transfer  the  path  of  integration  to  this 
surface ;  at  any  point  of  the  path  we  are  then  to  take  that  value 
of  the  root  which  belongs  to  this  point  on  the  surface. 

With  the  foregoing  provisions  we  are  prepared  to  answer  the 
question  whether  integral  (i)  is  expressible  in  terms  of  functions 
already  introduced.  As  a  matter  of  fact,  it  can  be  expressed  in 
this  wray  if  we  introduce,  by  means  of  the  equations  of  the 
previous  paragraph,  the  function  z  of  w  (and  £  of  o>)  defined  in 
that  paragraph,  as  the  variable  of  integration  ;  it  transforms 
accordingly  into  : 


=  i  log  ( —  iw  -f  Vi  —  w1). 

(According  to  the  stipulations  just  made  we  attached  the 
value  +  i  to  the  square  root  at  the  lower  limit.  We  therefore 
take  z  =  -\-  i  as  the  lower  limit  of  the  transformed  integral,  not 
z  =  —  i,  since  only  the  first  of  these  values,  viz.  z  —  -f  /',  belongs 
to  that  one  of  the  two  points  of  the  surface  lying  over  the  origin 
of  the  ay-plane  at  wrhich  the  square  root  has  the  value  +  i.) 

In  order,  therefore,  to  investigate  what  value  of  the  logarithm 
to  take  for  a  preassigned  path  of  integration,  or  how,  con 
versely,  to  select  the  path  of  integration  to  obtain  a  definite 
value  of  the  logarithm,  for  example,  the  principal  value,  we 
must  only  determine  how  the  paths  in  the  s-plane  and  on  the 
surface  over  the  o/-plane  correspond  to  each  other.  But  this  is 
easily  obtained  from  the  figures  of  the  previous  paragraph. 

If  we  limit  the  zopath  of  integration  to  the  first  sheet,  that 


362  V.    MANY-VALUED    ANALYTIC   FUNCTIONS 

of  z  remains  above  the  real  s-axis,  therefore  that  of  (—  iz]  to 
the  right  of  the  axis  of  pure  imaginaries  (—  iz),  and  the  loga 
rithm  of  (—  iz)  takes  on  its  principal  value.  We  will  designate 
the  corresponding  value  of  the  inverse  sine  as  its  principal 
value  in  the  first  sheet  of  the  RIEMANN'S  surface ;  its  real  part 
lies  between  —  77/2  and  -f  tr/2.  In  particular,  it  takes  on  con 
tinuously  increasing  the  real  values  from  —  ?r/2  to  +  77/2,  while 
w  continuously  increasing  takes  on  the  real  values  from  —  i  to 
H-i. 

Therefore  crossing  the  part  of  the  branch-cut  lying  to  the 
right  in  going  from  Al  to  Z>x,  or  from  B±  to  Clt  corresponds  in 
the  s-plane  to  crossing  the  half-axis  of  positive  reals,  and 
therefore  in  the  plane  of  (—iz)  to  crossing  the  negative  half- 
axis  of  pure  imaginaries  (—iz).  If  we  then  remain  in  the 
second  sheet,  without  again  crossing  a  cross-cut,  the  imaginary 
part  of  the  logarithm  remains  between  —iri/2  and  —  377-2/2, 
and  therefore  the  real  part  of  the  inverse  sine  between  77/2  and 
3  7T/2.  We  designate  this  value  as  the  "principal  value  of  the 
inverse  sine  in  the  second  sheet  of  the  RIEMANN'S  surface." 

Two  points  of  the  surface  which  are  situated  in  the  two 
sheets  one  vertically  above  the  other,  correspond  to  two  values 
of  z  whose  product  is  equal  to  i,  and  therefore  to  two  values  of 
(—  iz)  whose  product  is  equal  to  —  i.  The  sum  of  a  logarithm 
of  the  first  and  a  logarithm  of  the  second  of  these  values  is  an 
uneven  multiple  of  iri ;  the  sum  of  the  two  principal  values  of 
the  inverse  sine  is  exactly  equal  to  TT. 

But  if,  coming  from  the  first  sheet,  we  cross  the  part  of  the 
branch-cut  lying  to  the  left,  considerations  exactly  parallel  to 
the  foregoing  show  that  corresponding  to  this  in  the  plane  of 
(—  iz),  we  cross  the  positive  half-axis  of  pure  imaginaries  (—  iz), 
and  that  therefore  a  value  of  the  inverse  sine  is  obtained  in  this 
way  whose  real  part  lies  between  the  limits  —  7r/2  and  —  3  v/2. 


§62d.    THE   FUNCTION   Sin-1  w  363 

Let  us  go  from  the  first  sheet  over  the  part  of  the  branch-cut 
lying  to  the  right  into  the  second,  and  return  to  the  first  sheet 
over  the  part  of  the  branch-cut  lying  to  the  left ;  corresponding 
to  this  in  the  plane  of  (—12),  we  shall  then  start  in  the  half- 
plane  lying  to  the  right,  cross  the  negative  half-axis  of  pure 
imaginaries  into  the  half-plane  lying  to  the  left,  and  from  there 
cross  the  positive  half-axis  of  pure  imaginaries  again  into  the 
half-plane  lying  to  the  right.  But  this  is  making  a  circuit  about 
the  origin  in  this  plane  in  the  negative  sense ;  it  necessitates 
an  increase  of  the  logarithm  by  —  2  ?r/,  and  an  increase  of  the 
inverse  sine  by  2  TT. 

Such  a  closed  curve  upon  the  surface  can  be  transformed  by 
a  continuous  deformation  into  a  curve  which  surrounds  the 
point  at  infinity  in  one  "part  of  the  surface."  In  fact  the  resi 
due  at  the  point  at  infinity  of  the  function  of  w  to  be  integrated 
is  +  i  in  one  "  part "  of  the  surface  and  is  —  i  in  the  other 
"  part " ;  the  value  of  the  integral  taken  along  a  curve  which 
encircles  the  one  or  the  other  of  these  points  in  the  positive 
sense  is  accordingly  equal  to  ±  2  iri. 

It  is  now  possible  to  construct  the  most  general  path  upon 
the  surface  from  the  processes  heretofore  considered  and  their 
converses.  The  following  is  therefore  a  re"sum£  of  the  results : 

The  function  sin~l  w  defined  by  equation  (i)  is  an  infinitely 
many-valued  function.  Its  values  fall  into  fu>o  classes  corre 
sponding  to  the  two  values  of  the  square  root.  In  each  of  these 
classes  all  the  values  are  obtained  from  the  principal  value  by  the 
addition  or  subtraction  of  arbitrary  integral  multiples  of  2  TT.  The 
relation  between  the  principal  value  of  the  first  class  and  the  princi 
pal  value  of  tJie  second  class  is  that  their  sum  is  always  equal  to  IT. 

One  value  of  this  integral  is  found  to  be  equal  to  —  —  77  by 
introducing  17  as  the  variable  of  integration  by  means  of  equa- 


364  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

tion  (12),  §  62  c.  In  this  way  the  connection  with  the  EULER- 
IAN  relations  of  §  40  is  also  set  up  here.  Nevertheless  this 
way  of  considering  the  problem  to  its  ultimate  conclusion  would 
require  a  discussion  of  the  different  paths  of  integration.  But 
in  any  case  it  is  evident  that  integral  (i)  represents  the  com 
plete  inverse  of  the  sine  function  in  the  sense  that  it  has  for 
values  all  the  solutions  of  the  equation  sin/=  w  and  only  these. 


EXAMPLES 
1.   The  equation 

S*  7 

x  —  a 


/dx     _   i 
x2-a2  ~  2~# 


holds  when  a  is  real  and  (x  —  a}/(x  +  a)  is  positive.  If  we 
could  write  ia  instead  of  a  in  this  equation,  the  following  for 
mula  would  be  obtained  : 

tan-Y^  =  ^.  log  (*^*?U  const. 
* 


The  question  arises  whether,  now  that  the  logarithm  of  a  com 
plex  number  is  defined,  this  equation  is  not  actually  true. 

Since 

log(>  ±  id]  =  i  log  (X*  +  a1}  ±  (0  +  2  k*)i 

where  k  is  an  integer  and  6  the  numerically  least  angle  such 
that  cos  0  =  x/^/x"1  -f-  a2  and  sin  0  =  a/  V^2  +  #2,  we  have  at  once 


21 


where  /  is  an  integer,  and  this  does  differ  by  a  constant  from 


x 

any  value  of  tan"1)  - 

\a 


§63.    THE   FUNCTION    v^  365 

2.    The  standard  formula  connecting  the  logarithmic  and  inverse 
circular  functions  is 

tan-1  (*)  =  —  log  f  1±^|  ,    x  real. 
21     *  \i-ix)' 

Verify  this   formula   by  putting   jc  =  tanjy,   showing  that  it   is 
"  completely  "  true,  the  right-hand  side  reducing  to 

=    _  ,      (        2     }  = 


2  /        Vcos  _y  —  *  sin  yj      2  i 
where  k  is  any  integer. 

3.    Verify  the  formulas 

cos"1  x  =  —  i  log  (x  ±  i  V  i  —  x2),  sin"1  x  =  —  i  log  (ix  ±  Vi  —  x2), 
where  —  i  ^  x  ^  i,  each  of  which  is  also  "completely  "  true. 

§  63.   The  Function  \/z 

We  shall  find  no  particular  difficulty  in  the  study  of  the  «th 
root  of  z  after  the  investigation  of  the  special  case  of  the  square 
root  given  in  detail  in  the  last  paragraphs.  We  define  again  : 

I.    The  nth  root  of  a  complex  number  z 

(1)  ,=  & 

(n  a  positive  integer]  is    a  complex  number   s  which  satisfies  the 
equation 

(2)  r-« 

If  we  introduce 

(3)  rj  =  \OgZ 

as  the  independent  variable  as  in  §  58,  we  find  just  as  we  did 
before  that  : 


366  V.    MANY-VALUED   ANALYTIC   FUNCTIONS 

II.  We  obtain  all  the  pairs  of  corresponding  values  of  z  and  s 
which  satisfy  the  equation  (2),  if  we  put 

(4)  z  =  e1!,    s  =  e^ln 
and  consider  rj  as  the  independent  variable. 

III.  If  we  take  the  principal  value  ^for  log  z  in  (j),  we  obtain 
the  "principal  value  s0  of  the  nth  root  ""from  (4)  ;  it  is  characterized 
by  the  fact  that  its  amplitude  \\i  satisfies  the  conditions 

(5)  —  it ,ln  <  ^  <  T/n. 

All  the  other  values  of  the  logarithm  follow  from  its  principal 
value  by  the  addition  of  2  krci,  where  k  is  an  integer.  If  a 
is  the  smallest  positive  remainder  of  this  integer  according  to 
the  modulus  »,  we  obtain 

(6)  s  =  e«  -  J0 

by  substituting  77  =  ^0  -f-  2  kwi  in  (4) ;  in  this  equation  c  signifies 
(cf.  (3),  §  1 8)  the  definite  complex  number: 

/     \  —  2  7T     .      .     .        2   7T 

(7)  e  =  e  «  =  cos h  z  sin  — . 

n  n 

The  n  powers  of  this  number 

(8)  *  «  X,  ^  4*,.- •,.<-' 

are  all  different  from  each  other ;  for  suppose 

it  would  then  follow  that    ea~A  =  i, 

which  is  not  true.  It  follows  accordingly  that  in  addition  to  the 
principal  value  there  are  n  —  i  other  values  of  the  ;zth  root ; 
we  say : 

IV.  There  are  n  and  only  n  different  values  of  s  which  satisfy 
equation  (2]  for  each  value  of  the  complex  number  z  different  from 
O  and  oo . 


§63.    THE   FUNCTION    Vz  367 

Therefore  to  represent  the  ;/th  root  as  a  single-valued  func 
tion  of  position  on  a  surface,  we  need  only  n  sheets  of  the  in 
finitely  many-sheeted  surface  of  the  logarithm.  To  make  the 
function  also  continuous  on  the  surface,  the  ;/th  sheet  must  be 
attached  to  the  first  one :  to  do  this  the  final  border  of  the  #th 
sheet  must  penetrate  all  of  the  parts  of  the  surface  lying  under 
it,  in  order  to  reach  and  then  be  united  with  the  initial  border  of 
the  first  sheet  lying  lowest. 

We  can  best  obtain  an  idea  of  this  surface  by  thinking  of  its 
gradual  formation.  This  is  done  as  in  §  59  for  the  special  case 
where  n  =  2  ;  we  have  now  only  to  let  the  moving  radius  make 


FIG.  36 

n  circuits  instead  of  2,  and  immediately  after  completing  the  rth 
circuit  pierce  the  parts  of  the  surface  lying  under  this  radius 
and  then  be  combined  with  the  initial  border.  For  ;/  =  4, 
Fig.  36  represents  a  section  through  the  surface  perpendicular 
to  the  half-axis  of  negative  real  numbers,  looking  at  the  section 
from  the  origin.  The  origin  is  a  branch-point  of  the  surface  of 
order  (#—  i);  transforming  from  the  plane  to  the  sphere 
shows  the  point  oo  also  to  be  a  branch-point  of  order  (;/  —  i). 

V.  The  connectivity  of  this  surface  is  the  same  as  that  of  the 
sphere  even  in  the  general  case  when  n  is  arbitrary.  This  may  be 
shown  by  any  of  the  methods  spoken  of  in  §  60.  If  we  wish  to 
make  a  continuous  deformation  of  the  surface,  we  must  think  of 
the  sheet  farthest  inside  as  drawn  out  of  the  one  next  to  it,  and 
then  think  of  the  sphere  thus  generated  from  these  two  sheets  as 


368 


V.    MANY-VALUED   ANALYTIC   FUNCTIONS 


drawn  out  of  the  sheet  third  from  the  inside,  etc.     Let  us  make 

a  provisional  dissection  of  the  surface  with  the  understanding 

that  it  be  subsequently  combined  ; 
we  now  deform  each  individual 
sheet  according  to  the  process 
given  at  the  end  of  §  60  until  the 
angle  at  the  origin  is  reduced  to 
2  TT/#,  and  then  place  the  sheets 
adjacent  to  each  other.  It  is 
scarcely  necessary  to  mention  that 
the  sphere  arranged  in  this  way 
can  be  mapped  conformally  upon 

the  ^-sheeted  surface  by  the  equation  sn  =  z. 

The  functions  which   are  regular  on    this    surface  with    the 

exception  of  certain  poles  are  rational  functions  of  s  =  -\/z  and 

may  be  treated  as  in  §  60  a. 
The  discussion  of  the  function 


FIG.  37 


Vn'z  —  a 
nr*1 


only  apparently  more  general,  may  be  referred  to  that  of  Vz,  as 
was  done  for  n  =  2  in  §  62.     On  the  other  hand  the  function 


A/0  -  a)(z  -  //),     (n  >  2) 

belongs  to  another  class  of  irrationalities ;  it  has  a  branch-point 
at  infinity  in  addition  to  those  at  a  and  b. 

§  64.    The  Equation  s2  =  i  -  z3 

As  an  example  of  a  somewhat  less  simple  algebraic  relation  ex 
hibiting  the  dependence  between  z  and  s,  we  call  attention  to 
the  equation : 


§  64.    THE   EQUATION   j*  =  I  -  s>  369 

The  equation  shows  that  s  is  a  double-valued  function  of  z ;  the 
factors :  27rt 

(2)  (i-s3)=(i-s)(€-c)(e2-,),      *«,« 

show  that  j-  changes  its  sign  when  z  makes  a  circuit  about  one  of 
the  points  i,  e,  e2,  and  that  therefore  these  points  are  branch^ 
points  in  the  s-plane.  In  addition  to  this  the  point  z  =  oo  is  also 
a  branch-point  as  is  shown  by  the  development : 

fr--jpt_£jrwt.f.  •••. 

To  separate  out  a  single-valued  branch  of  s,  we  connect  these 
four  points  by  cuts  in  such  a  way  that  it  is  not  possible  to  make 
a  circuit  around  any  one  of  them  without  crossing  a  cut.  This 
can  be  done  symmetrically  by  drawing  three  cuts  from  the  three 
points  to  infinity  in  such  a  way  that  when  prolonged  in  the 
opposite  direction  they  pass  through  the  origin.  To  obtain  now 
a  surface  on  which  s  can  be  represented  as  a  single-valued  and 
continuous  function  of  position,  we  take  two  s-planes,  each  treated 
in  this  way,  and  fasten  them  together  crosswise  along  the  cuts. 


Conversely,  z  =  \(i  —  s)(i  -f-  s) 

is  a  triple-valued  function  of  s.  If  s  encircles  one  of  the  points 
-f  i  or  —  i  of  its  plane  in  the  positive  sense,  e  enters  each  time 
as  a  factor  of  z  ;  these  two  points  are  therefore  branch-points  in 
the  .r-plane.  In  addition  j-  =  oo  is  a  branch-point.  We  must 
therefore  connect  one  of  these  three  branch-points  by  cuts  with 
the  other  two ;  this  is  obtained  symmetrically  when  a  cut  is 
made  in  the  j--plane  along  the  real  .r-axis  with  the  exception  of 
the  part  between  --  i  and  -f  i.  We  then  take  three  j-planes 
cut  in  this  way  and  connect  them  along  the  cuts.  Let  us  define 
the  sheets  in  such  a  way  that,  for  s  =  o,  z  =  i  in  the  first  sheet, 
z  =  e  in  the  second  sheet  and  z  =  e2  in  the  third  sheet ;  a  posi 
tive  circuit  about  each  of  the  two  branch-points  lying  on  the 


3/0  V.    MANY- VALUED   ANALYTIC   FUNCTIONS 

finite  part  of  the  surface  therefore  leads  from  the  first  sheet  into 
the  second,  from  this  one  into  the  third  and  from  this  one  again 
into  the  first  sheet ;  accordingly  we  must  connect  the  positive 

f  i  ) 
half-plane  of  the  \  2  [  sheet  with  the  negative  half-plane  of  the 

\  3  j-  sheet  along  the  cut  from  oo  to  —  i,  but  the  positive  half- 

[i] 
plane  of  the   \  2  \  sheet  connects  with  the  negative  one  of  the 


1  }•    sheet  along  the  cut  from   i  to  oo.      We  thus  have  two 

2  j 

branch-cuts  from  the  point  at  infinity  along  which  the  sheets 
are  connected  differently ;  a  check  on  these  results  is  the  fact 
that  one  circuit  about  this  point  in  the  positive  sense  (that  is, 
so  that  the  point  lies  to  the  left)  transfers  us  from  the  first  to 
the  second  sheet  as  it  should  be.  (The  development 


holds  in  the  neighborhood  of  s  =  oo ;  and  if  we  encircle  s  =  oo 
in  the  positive  sense,  e2  enters  as  a  factor  of  sl/s  and  conse 
quently  e  as  a  factor  of  each  term  of  the  given  development.) 

The  above  two-sheeted  surface  over  the  s-plane  and  this 
three-sheeted  surface  over  the  .f-plane  are  mapped  by  equation 
(i)  reversibly  and  uniquely  and  in  general  conformally  upon 
each  other  (that  is,  excepting  the  branch-points  of  the  two  sur 
faces  and  their  images).  To  carry  out  this  representation  in  detail 
we  first  determine  what  lines  of  each  surface  correspond  to  the 
branch-cuts  of  the  other  surface.  For  this  purpose  let  us  put 

z  =  x  +  />',    s  =  u  +  tv, 

and  separate  equation  (i)  into  its  real  and  imaginary  parts;  we 
obtain  : 

(3)  «•-*»=!-. 

(4)  2^  =  -3. 


64.    THE   EQUATION   /2  =  I  - 


371 


Hence  the  line  u  =  o  (in  the  three  sheets  of  the  surface  over 
the  .r-plane)  corresponds  to  the  branch-cuts : 


and  the  lines 


.}, 


X  <  O, 


x>o, 


_>-  =  -W3J 

in  the  two  sheets  of  the  surface  over  the  s-plane,  correspond 
to  the  branch-cuts  : 

v  =  o,    u  <  —  i   and  v  =  o,    u  >  i. 

Construction  of  these  lines  divides   each  of   the  two  surfaces 
into  six  parts ;  these  parts  correspond  to  each  other  as  shown 


s-plane-1 
=B  A= 


s-p/ane-ll  s-p/ane-lll 


z-p/ane-ll 


FIG.  38 


in  Fig.  38.  To  further  determine  this  correspondence,  we 
must  distinguish  also  between  the  two  sheets  of  the  surface 
over  the  s-plane  (as  was  unnecessary  above)  ;  this  is  done  by 
arranging  that  s  should  =  i  at  zero  of  the  first  sheet,  and  that 


372 


V.    MANY-VALUED   ANALYTIC   FUNCTIONS 


s  should  =  —  i  at  zero  of  the  second  sheet.  Thus,  for  example, 
the  region  A  is  defined  from  the  fact  that  it  contains  the  points 
(z  =  o,  s=i)  and  (z=i,  J  =  o);  and  B  is  likewise  defined,  con 
taining  (0  =  0,  s  =  —  i)  and  (3=  i,  j  =  o),  etc. 

To  investigate  still  further  the  mapping  of  the  region  Aa  upon 
the  region  At,  we  find  from  a  study  of  the  formulas  (3)  and  (4) 
that  the  following  lines  of  the  two  regions  correspond  : 

u2  —  zP=i,  u  >  o,  v  >  o  •••  x  +  }'^/3  =  o,  y  <  o 
&2—  ^2—  i,  u  >  o,  z;  <  o  •••  x  —  7A/3  =  o,  jy  >  o 
z>  =  o,         o  <  //  <  i  —  y  =  o,  o  <  x  <  i. 

We  thus  obtain  just 
four  subregions  which 
correspond  to  each 
other  as  shown  in 

Fig-  39- 

A     study    of     the 

curves  which  corre 
spond  to  the  parallels 
to  the  axes  in  each  of 
the  planes  would  be 


s-p/ane-l 


FIG.  39 


z-pfane-1 


of  no  aid  in  obtaining  further  details  here.    On  the  contrary  we 
find  from  equations  (3)  and  (4)  that  the  hyperbolas  of  the  ^-plane  : 

u"1  —  v1  =  C\,    2  uv  =  C2, 

correspond  to  curves  of  the  s-plane  whose  properties  are  ob 
tained  from  their  equations  in  polar  coordinates : 


cos  3  <£  =  i  —  Cl}   ps  sin  3  <£  =  —  C2 


§65.    WEIERSTRASS'S   DEVELOPMENT   IN   A   PRODUCT      373 

EXAMPLES 

1.  Show,    for   w  =  z~  —  i ,    that    as    w    describes    the    circle 
|  w  \  =  k,  the  two  corresponding  positions  of  z  each  describe  the 
Cassinian    oval  Pl  •  p2  =  k  (Pl,  p2  being  the  distances  from  the 
points  ±  i).     Trace  the  ovals  for  different  values  of  k. 

2.  If  7t'  =  2  z  +  z~,  show  that  the  circle  \  z  \  =  i    corresponds 
to  a  cardioid  in  the  plane  of  w. 

3.  If   («'+i)2  =  4/s,  the    unit   circle   in    the   w-plane   corre- 

n 
spends  to  a  parabola  r  cos2  -  =  i   in  the  s-plane,  and  the  inside 

of  the  circle  to  the  outside  of  the  parabola. 

4.  Show,  for  the  transformation  w  =  \  (z  —  ta)/(z  +  id)  J2,  that 
the  upper  half  of  the  w-plane  may  be  made  to  correspond  to 
the  interior  of  a  certain  semi-circle  in  the  z-plane. 

5.  If  K>  =  azm  +  bzn,  where  ;;/,  ;/  are  positive  integers  and  a,  b 
real,  show    that  as   z  describes  the  unit  circle,  w  describes  a 
hypocycloid  or  an  epicycloid. 

6.  Discuss  the  mapping  of  parallels  to  the  s-axes  by  means  of 
cot  0. 

7.  Show  that  a  cut  along  a   complete    hyperbola    separates 
branches  of  sin"1  w. 

8.  If  w  =  cosz,  2  w  =  rj  +  -    where  77  —  e".     Hence  when  z 

•n 

moves  horizontally  or  vertically  determine  the  map  on  the  ^-plane 
and  then  on  the  ay-plane. 

§  65.     Transition  from  MlTTAG-LEFFLER'S  Division  into  Partial 
Fractions  to  WEIERSTRASS'S  Development  in  a  Product 

Suppose  we  have  a  given  function  of  the  kind  considered  in 
§  51,  all  of  whose  poles  are  simple  and  all  of  whose  residues 


374  v-    MANY-  VALUED   ANALYTIC   FUNCTIONS 

=  i,  and  which  consequently  may  be  represented  by  a  series  of 
the  form  : 

(i)  +(i)ssl-J—+a    +  a1,g+  -avk 


we  can  then  show  (as  the  converse  of  Theorem  II,  §  46)  that  this 
function  is  the  logarithmic  derivative  of  a  transcendental  integral 
f  unction  f(z)  ;  that  is,  that 


According  to  VI,  §  35,    I    </>(z)  dz  is  regular  in  every  simply  con- 

c/O 

nected  domain  which  contains  none  of  the  points  av  in  its  in 
terior  ;  if  z  encircles  one  of  the  points  avJ  this  integral  is  increased 
by  2  TT/.  Consequently  if  b  is  not  one  of  the  points  avt 


exp 


is  a  regular  function  over  the  whole  plane  apart  from  the  points 
av ;  in  the  neighborhood  of  av  it  is  equal  to  the  product  of  z  —  av 
by  a  regular  function.  It  can  therefore  be  made  a  regular 
function  in  the  whole  plane,  that  is,  a  transcendental  integral 
function,  by  assigning  to  it  the  value  zero  at  the  points  av. 
Thus  <£(z)  is  the  logarithmic  derivative  of  this  transcendental 
integral  function. 

On  account  of  its  uniform  convergence,  the  series  (i)  may  be 
integrated  term  by  term  along  an  arbitrary  path  which  does  not 
contain  any  of  the  points  av.  Without  loss  of  generality1*  we 
may  suppose  that  zero  is  not  one  of  the  av ;  we  can  then  use 
zero  as  the  lower  limit  of  the  integral  and  so  obtain  the  follow 
ing  series  which  is  absolutely  and,  in  the  same  domain  as  (i), 

*  If  zero  belongs  to  the  av  we  need  only  to  investigate  <}>(z)  —  i/z  instead  of  $(z). 


§  65.    WEIERSTRASS'S  DEVELOPMENT  IN  A   PRODUCT      375 
uniformly  convergent  : 
(3) 


Since  the  exponential  function  is  a  continuous  function  of  its 
argument  (A.  A.  7,  §  50)  the  lemma  that 

(4)  exp  (  lim  sn)  =  lim  (exp  sn) 

n  =  oo  n  =  oo 

is  true  (provided  lim  sn  exists)  ;  from  it  and  from  the  definition 

n  =  » 

of  the  infinite  series  and  of  the  infinite  product,  it  follows  that 

GOO  v  -X 

5^1=11^1 
=  1  /  V  =  l 

that  is  : 

°°^  oe 

I.     When  the  series  Zj  uv  converges,  the  product  JJ  e^  also  con- 

V  =  l  l'=l 

verges,  and  in  fact  to  a  value  different  from  zero  in  the  limit. 

Consequently  we  can  deduce  from  equation  (3)  the  following  : 


II.  The  transcendental  integral  function  f(z)  for  which  the 
points  av  are  simple  zeros  may  be  represented  analytically  in  the 
form  of  the  infinite  product  (6),  provided  that  the  points  av  satisfy 
the  conditions  of  §  57  a?id  that  the  coefficients  av(t  are  determined 
according  to  the  rules  given  there. 

If  now  we  have  given  any  transcendental  integral  function 
F(z)  for  which  also  the  points  av  are  simple  zeros,  the  quotient 
^(z)//(z)  w^l  be  a  function  regular  over  the  whole  plane,  and 
consequently  a  transcendental  integral  function  E(z)  which  in 
addition  is  nowhere  zero.  The  logarithm  of  such  a  function  is 


3/6      V.  MANY- VALUED  ANALYTIC  FUNCTIONS 

also  regular  over  the  whole  plane  (cf.  X,  §  38) ;  consequently, 
we  have 

(7)  £(*)**'** 

in  which  g(z)  is  also  a  transcendental  integral  function.     Hence 
the  theorem  : 

III.  The  most  general  transcendental  integral functio  n  for  which 
the  points  av  are  simple  zeros  is  represented  in  the  form 

(8)  F(z)  =/(*>•<*> 

in  which  f(z)  is  the  product  (6),  and  g(z)  is  any  transcendental 
integral  function. 

As  an  illustration  of  Theorem  II  we  cite  the  following  two 
product  forms  of  the  sine  which  are  obtained  from  the  develop 
ments  of  the  cotangent  in  partial  fractions,  (2)  and  (18),  §  52  : 

(9)  sin  TTZ  =  TTZ  -  IT  '  (  (i  - 
and 

(10)  sin7r(tf +  £)  =  sin(7 
in  particular  for  a  =  1/2  : 

-4-0 

(11)  COS7TS=I 


2  V  —   I 


The  accent  on  the  product  sign  in  (9)  has  a  meaning  analogous 
to  that  given  earlier  to  the  accent  on  the  summation  sign. 

If  in  the  products  (9)  and  (n)  we  take  together  in  pairs  those 
factors  which  belong  to  oppositely  equal  values  of  v  and  2  v  —  i 
respectively,  we  obtain  the  elementary  product-forms  for  these 
functions  (A.  A.  §  83)*. 

*That  is,  sins-  mf[l  i —  V  cosz  =  f[(i— ^ V  —  S.E.R. 

' 


MISCELLANEOUS    EXAMPLES  377 

EXAMPLE 

Write  down  an  infinite  product  which  defines  a  transcendental 
integral  function  of  z  having  simple  roots  in  the  points 

z  =  n  +  //«,    n=i,  2,  •••  , 

but  not  vanishing  elsewhere.     Prove  that  the  product  has  the 
desired  property. 

MISCELLANEOUS   EXAMPLES 

1.  Show  that  if  0  is  real  and  sin  0  sin  <£  =  i,  then 

<£  =  (k  4-  l/2)ir  +  /  •  log  COt  4-0&7T  +  0) 

where  k  is  any  even  or  any  odd  integer,  according  as  sin  0  is 
positive  or  negative.     Cf.  examples  following  §  40  and  §  62  d. 

2.  If   a  cos  0  -f  b  sin  0  +  c  =  o,    where    #  ,  £,  c   are    real  and 
f2  >  a*  +  £2,  then 


.  -i       f  IH  +  V^-a2-^) 

^  =  ;;/TT  -f  tt  ±  I  log     {  L 

V«2  +  ^2 

where  m  is  any  odd  or  any  even  integer,  according  as  c  is  posi 
tive  or  negative,  and  a  is  the  least  angle  whose  cosine  and  sine 
are  tfV 


3.  Show  that  if  .#  is  real,  then 

—  exp  {(a?  +  M)x\  =(a  +  /^)^«+*>',    fexp  {(*  4-  ^)jej  ^ 
M»X^  »/ 

_  exp  (a  +  iti)x 
(a  +  tt} 

4.  Prove  that,  for  a  >  o,     |    exp  |  —  (a  +  #).*j*/.r=  —  ^—  r. 

»/D  ^7  +  W 

5.  Determine  the  number  and  the  approximate  positions  of 
the  roots  of  the  equation  tan  z  =  az,  where  a  is  real, 


3/8      V.  MANY- VALUED  ANALYTIC  FUNCTIONS 

It  is  easily  shown  that  this  equation  has  infinitely  many  real 
roots.     Next  let  z  =x  -\-iy  and  equate  real  and  imaginary  parts. 

(sin  2  ;r)/(cos  2  x  -f  cosh  2  y)  =  ax, 
(sinh  2  y)/(cos  2  x  +  cosh  2  y)  =  ay, 

and  therefore,  if  x  and  jy  are  not  zero,  we  have 
(sin  2  x)/2  x  =  (sinh  2  _y)/2  _y. 

But  this  is  impossible,  since  the  left-hand  side  is  numerically 
less,  and  the  right-hand  side  numerically  greater,  than  unity.  It 
follows  that  x  —  o  or  y  =  o.  But  if  y  =  o,  we  come  back  to  the 
real  roots  of  the  equation.  If  x  =  o,  tanh  y  =  ay.  It  may  be 
shown  graphically  that  this  equation  has  no  real  root  other  than 
zero  if  a  ^  o  or  a  ^  i,  and  two  such  roots  if  o  <  a  <  i.  Thus 
there  are  two  purely  imaginary  roots  if  o  <  a  <  i  ;  otherwise 
all  the  roots  are  real. 

6.  The  equation  tan  z  =  az  +  b,  a  and  b  real  and  b  3=  o,  has 
no  complex  roots  if  a  <^  o.     If  a  >  o  the  real  parts  of  all  the 
complex  roots  are  numerically  greater  than  |  b/2  a  \ .    Prove. 

7.  The  equation  tan  z  =  a/z,  a  real,  has  no  complex  roots  but 
has  one  purely  imaginary  root  if  a  <  o.    Prove. 

8.  Discuss  the  transformation  z  =  <:  •  cosh  (irw/a),  showing  in 
particular  that  the  whole  s-plane  corresponds  to  any  one  of  an 
infinite  number  of   strips  in  the    w-plane  each  parallel  to  the 
^-axis  and  of  breadth  2  a.     Show  also  that  to  the  line  u  =  UQ 
corresponds  the  ellipse 


,          TTZ/n 

rcosh   - 
V  a 


c  sin 


\  a 


and  that  for  different  values  of  u0  these  ellipses  form  a  confocal 
system  ;  and  that  the  lines  v  —  VQ  correspond  to  the  associated 


MISCELLANEOUS   EXAMPLES  379 

system  of  confocal  hyperbolas.  Trace  the  variation  of  z  as  w 
describes  the  whole  of  a  line  it  =  //0  or  v  =  z-0.  How  does  w 
vary  as  z  describes  the  degenerate  ellipse  and  hyperbola  formed 
by  the  segment  between  the  foci  of  the  confocal  system  and  the 
remaining  segments  of  the  axis  of  x  ? 

9.    Verify  that  the   transformation  z  =  c  cosh  (vw/a)  can  be 
compounded  from  the  transformations 

z  =  cz^    %  =  ^  (z2  +  i  /%)?    z.,  =  c  exp  (TTW/O). 

10.  Discuss    similarly    the    transformation  z  =  c  tanh  (irw/a), 
showing  that  to  the  line's  u  =  //„  correspond  the  coaxial  circles 

\x-c  coth  (iruo/a) j2  +  /  =  c1  cosech2  (irt/0/a), 

and  to  the  lines  v  —  z»0  correspond  the  orthogonal  system  of 
coaxial  circles. 

11.  Discuss  the  transformation 

fw  —  a  + 


z=iog  I 

showing  that  the  straight  lines  for  which  x  and  y  are  constant 
correspond  to  sets  of  confocal  ellipses  and  hyperbolas  whose 
foci  are  the  points  w  =  a  and  w  =  b. 


Here      ^/(w  -  a)  +  V(w  -  b}  =  V(£  -  a)  exp  (x  +  /» 


—  of)  exp  (  —  A*  —  / 
and  it  is  readily  found  that 

\u>  —  a\  +  \w  —  l>\  =  \l>  —  a\-  cosh  2  x, 
\w  —  a\  —  \w—b\=   b  —  a  \  •  cos  2  y. 

12.    Prove  that  if  neither  a  nor  b  is  real  then 

*  ~  L 
a  —  b 


380  V.    MANY- VALUED   ANALYTIC   FUNCTIONS 

each  logarithm  having  its  principal  value.  Verify  the  result  if 
a  =  ci,  b  =  —  d  where  c  is  positive.  Discuss  the  cases  where  a 
or  b  or  both  are  real  and  negative. 

13.    Show  that  if  «  and  ft  are  real,  and  (3  >  o, 


What  is  the  value  of  the  integral  when  (3  <  o  ? 

14.  If  an  algebraic  plane  curve  has  a  double  point  with  dis 
tinct  tangents  neither  of  which  is  vertical,  what  can  be  said  of 
the  corresponding  RIEMANN'S  surface  ? 

15.  Of  a  certain  f  unction  /(z)  I  know  that  it  is  single-valued 
and  regular  in  the   region    of   the    s-plane    lying   between   the 

ellipses  ^        2  ^      y 

—  H  ----  —  *>    --  1  --  T  —  r> 
49  25      36 

and  that  along  the  arc  of  the  circle  of  radius  4,  with  its  center 
at  the  point  z  =  o,  which  lies  in  the  first  quadrant  f(z)  has  the 
value  3  •  5  —  8  •  3  /.  What  can  you  say  about  f(z]  ? 

16.  Find  all  the  values  of  tan"1^  -f  /)  to  three  figures. 

17.  The  function  of  the  real  variable  x  defined  by 


(where  7(V)  denotes  the  imaginary  part  of  »)  is  equal  to/  when 
x  is  positive,  and  equal  to  q  when  x  is  negative. 

18.    The  function  of  x  defined  by 


is  equal  to  /  for  x  >  i  ,  to  ^  for  o  <  jc  <  i  ,  to  r  for  ^  <  o. 

19.  Draw  the  graph  of  the  function  /(log  a;)  of  the  real  varia 
ble  x.  (The  graph  consists  of  the  positive  halves  of  the  lines 
y=  2  kir  and  the  negative  halves  of  the  lines  y  =(2  k  +  iV.) 


MISCELLANEOUS   EXAMPLES  381 


20.  Show  that  exp  (i  +  i)z  =  V  22"  .  exp  [  -  mri  )  •  —  . 

VI       /   «! 

21.  Expand  cos  2  cosh  2  in  powers  of  z. 

We  have  cos  z  cosh  z  —  /  sin  z  sinh  z  =  cos  (i  +  i)z 


Similarly  cos  z  cosh  z  +  i  sin  z  sinh  z  =  cos  ( i  —i)z 

_IV,i»5       _[_  \n>          >          f          I 

2V  \     4    *  7    n\ 

Hence  cos  z  cosh  z  =  -  V  22" J  i  +(—  i)nf  •  cos-  «TT  •  — 
"  4         »! 


22.    Expand      sin  z  sinh  2,    sin  z  cosh  2,    cos  z  sinh  z   each    in 
powers  of  z. 


CHAPTER   VI 

GENERAL  THEORY  OF  FUNCTIONS 
§  66.    The  Principle  of  Analytic  Continuation 

WE  have  already  investigated  a  series  of  many-valued  func 
tions  of  a  complex  variable  in  the  previous  chapter  ;  the  question 
of  prime  importance  in  this  connection  is  the  following:  When 
several  values  of  one  complex  variable  are  associated  with  each 
value  of  another,  under  what  conditions  are  these  first  values, 
taken  together,  to  be  regarded  as  a  many-valued  function  of  the 
latter  (and  not  as  different  single-valued  functions)  ?  In  the  in 
vestigation  of  this  question  we  begin  with  the  following  con 
siderations  : 

Let  a  bounded  domain  S  and  a  function  of  z,  regular  in  this 
domain,  be  given  in  the  plane  (or  on  the  sphere).  We  consider 
then  a  domain  S'  of  which  S  is  a  part,  and  inquire  whether  a 
function  exists  which  is  regular  and,  by  definition,  single-valued 
everywhere  inside  of  S'  and  which  is  identical  with  the  first 
named  function  inside  of  S.  (That  only  one  such  function  can 
exist  in  any  case,  when  one  exists  at  all,  follows  from  theorem 
VII,  §39-) 

I.  If  such  a  function  is  found  then  we  say,  according  to  WEIER- 
STRASS:  we  have  continued  the  given  function  analytically  beyond 
the  given  domain  for  which  it  is  defined. 

The  question  concerning  the  existence  of  such  a  function  may 
be  regarded  as  belonging  to  the  subject  of  linear  partial  differ 
ential  equations.  The  real  and  the  imaginary  parts  of  a  regular 

382 


§  66.    THE   PRINCIPLE  OF  ANALYTIC  CONTINUATION      383 

function  of  a  complex  argument  satisfy,  as  we  know,  the  CAUCHY- 
RIEMANN  differential  equations  ;  another  formulation  of  the 
problem  is  therefore  the  following  :  Given  the  values  of  two 
functions  //,  v  along  a  line  L  (a  piece  of  the  boundary  of  the 
original  domains)  ;  we  desire  to  find  two  functions  u,  v  which  sat 
isfy  the  differential  equations  : 

du  _  dv        dv  _      du 

Cx      cy        ex  dy 

in  the  neighborhood  of  this  curve  and  which  reduce  to  u,  v 
respectively  along  this  curve.  But  this  formulation  of  the  prob 
lem  leads  into  difficulties  when  we  attempt  to  state  precisely 
what  continuity  properties  are  presupposed  for  the  curve  L  and 
the  assigned  values  along  Z,  and  what  properties  of  this  kind  we 
may  require  of  the  functions  to  be  determined.  On  this  account 
the  problem  is  not  discussed  here  from  this  standpoint,  but  we 
use,  as  did  WEIERSTRASS,  the  development  of  the  regular  func 
tions  in  power  series. 

Let  a  regular  function  f(z)  be  defined  in  a  domain  S,  and  let 
a  be  an  inner  point  of  this  domain.     The  TAYLOR'S  series 


then  converges  (III,  §  37)  at  any  rate  inside  of  the  largest  circle 
F  with  center  a  which  belongs  entirely  to  the  domain  S,  and  in 
fact  converges  to/(z).  But  it  is  altogether  possible  that  it  con 
verges  outside  of  T  and  imide  of  a  circle  T'  concentric  with  T. 
The  surface  of  this  circle  T'  has  at  least  one  continuous  domain 
5  in  common  with  the  given  domain  £  of  which  the  surface  of 
T  is  a  part  ;  it  is  also  possible  (cf.  Fig.  40)  that  it  has  in  com 
mon  with  S  another  or  several  other  domains  3'  which  are  not 
connected  with  2,  ;  for  these  however  the  following  theorems  do 
not  hold.  But  inside  of  2  the  value  of  series  (i)  is,  according 


VI.  GENERAL  THEORY  OF  FUNCTIONS 


FIG.  40 


to  V,  §  38,  a  single-valued,  reg 
ular  function  of  z  which  may 
be  designated  provisionally  by 
<£  (z).  The  difference 

*(*}-/(*) 

is  therefore  regular  everywhere 
inside  of  2J  and  is  everywhere 
=  o  in  a  part  of  2,  viz.  inside  of 
P.  Hence  it  is  zero  in  the  whole 
domain  2  according  to  VII,  §  39  ; 
that  is,  we  have  the  theorem  : 


II.  When  series  (7)  converges  also  at  points  which  do  not  belong 
to  the  original  domain  for  which  the  function  f(z)  is  defined,  then  the 
two  functions  coincide  in  the  whole  continuous  domain  25,  which  is 
common  to  the  domains  defining  the  function  and  the  series  and 
which  contains  the  point  a. 

Definition : 

III.  Series  (/)  represents  an  "  analytic  continuation  "  of  the  given 
"  element  of 'the  function"  f '(z)  in  all  parts  of  its  domain  of  conver 
gence  Si  not  belonging  to  2 ;    the  domain  for  which  this  function 
was  defined,  originally  limited  to  S,  is  in  this  way  enlarged. 

IV.  All  the  elements  obtained  from  a  given  element  of  the  func 
tion  by  repeated  analytic  continuation  together  constitute  an  analytic 
function* 

The  many-valued  functions  investigated  in  the  previous 
chapter  satisfy  this  definition  as  is  easily  shown.  We  can  go 

*  The  analytic  function  is  thus  defined  by  a  power  series,  whose  radius  of  con 
vergence  is  not  zero,  together  with  all  possible  continuations  of  that  series.  Cf. 
HARKNESS  AND  MORLEY,  Introduction,  etc.,  pp.  154,  314;  OSGOOD,  Lehrbuch, 
Vol.  i,  pp.  89,  189.  —  S.  E.  R. 


§  6;.    GENERAL   CONSTRUCTION  OF   RIEMANN'S   SURFACE    385 

from  one  branch  of  the  function  to  any  other  branch  by  analytic 
continuation ;  but  such  continuation  cannot  lead  to  values 
other  than  those  that  are  each  time  under  consideration.  This 
latter  statement  follows  from  the  following  general  theorem  : 

V.  If  a  function  f(z)  is  by  definition  regular  in  a  domain  S 
and  if  it  satisfies  an  equation : 

G(z,f(z),f'(z»=o 

at  all  points  of  this  domain,  where  G  is  understood  to  be  a  rational 
integral  function,  then  the  same  equation  holds  for  all  analytic  con 
tinuations  off(z). 

To  prove  this  theorem  we  develop  G  in  powers  of  z  —  a  ; 
since  this  development  is  by  hypothesis  zero  everywhere  inside 
of  2,  G  must  be  zero  everywhere  inside  of  Sl  according  to 
VII,  §  39- 

The  analytic  continuation  of  the  integral  of  a  single-valued 
function  is  particularly  simple ;  such  an  integral  is  defined  at 
present  as  a  single-valued  function  of  its  upper  limit,  in  a  simply 
connected  domain  which  contains  the  lower  limit  but  no  singular 
point  of  the  function  to  be  integrated ;  that  is,  while  the  path  of 
integration  remains  entirely  in  this  domain  (VI,  §  35),  If  the 
path  of  integration  then  reaches  beyond  this  domain,  we  obtain 
an  analytic  continuation  of  the  element  of  the  function  first 
defined :  and  different  continuations  of  this  kind  lead  to  differ 
ent  values  of  the  function  when  the  path  of  integration  con 
sidered  encloses  a  singular  point  at  which  the  residue  is  not 
zero.  Examples  of  this  are  found  in  §§  56,  57  a,  62  d. 

§  67.     General  Construction  of  the  RiEMANN'S  Surface  determined 
by  an  Analytic  Function 

As  in  the  previous  paragraph,  let  an  element  of  the  function 
f(z)  be  given  in  a  domain  Si ;  suppose  we  have  found  a  con- 


386 


VI.    GENERAL  THEORY   OF   FUNCTIONS 


tinuation  fi(z)  of  f(z)  in  a  domain  S2  which  has  a  continuous 
domain  Si  in  common  with  Si  ;  then  suppose  a  second  continua 
tion  /^(z)  in  a  domain  ^3  which  has  a  continuous  domain  S2  in 
common  with  (^  +  S2  —  Si)  ;  then  a  third  continuation,  etc.  ; 
finally  an  nth  continuation  in  a  domain  Sn+i  which  has  a  con 
tinuous  domain  S  in  common  with 


It  is  now  possible  that  Sn+i  has  in  common  w'ith  Si  a  domain 
SB+1  which  is  not  connected  with  Si.  (Fig.  40  shows  this  pos 

sibility  for  72=  i,  Fig.  41  shows 
it  for  »  =  5.)  In  this  domain, 
therefore,  two  elements  of  the 
function  are  defined,  viz.  f  and 
fn  ;  but  we  have  yet  no  basis 
for  the  statement  that  these 
elements  must  always  be  the 
same.  We  have  accordingly 
two  cases  to  dispose  of. 

I.     When    all   the   continua- 
tMG-4I  tions  which  are  obtainable  di 

rectly  or  indirectly  from  a  given  element  of  the  function,  always 
furnish  the  same  values  of  the  function  for  the  same  values  of  the 
argument,  we  say  :*the  element  of  the  function  first  given  generates 
a  single-valued  analytic  function. 

But  when  that  is  not  the  case,  the  existing  relations  are  made 
clear  by  the  following  geometrical  representation.  We  think 
of  the  defining  domain  of  the  function  as  increasing  step  by  step 
by  adding  in  turn  to  the  original  domain  Si,  first  S2  —  Si,  then 
*S3  —  S2,  etc.  When  finally  Sn+l  —  Sn  has  a  part  2n+i  extending 
over  a  part  of  the  original  domain,  as  in  Figs.  40  and  41,  and 
^  coincides  with  /in  this  part,  then  we  add,  not  the  whole 


§  67.   GENERAL  CONSTRUCTION  OF  RIEMANN'S  SURFACE      387 


of  ^"n+1  —  2re  but  only  SnJ.i  —  2B  —  2n+i  5  removing  the  bound 
ing  curve  between  the  newly  added  piece  and  2no-i,  we  have  a 
doubly  connected  domain  (eventually  multiply  connected).  This 
domain  is  momentarily  the  defining  domain  of  the  function  ;  it  is  by 
definition  single-valued  in 
this  domain.  But  when 
fn  does  not  coincide 
with  f  in  2n+1,  then  we 
add  on  all  of  Sn+l  —  2n 
to  the  existing  domain 
which  may  be  regarded 
as  a  material,  flat  sheet. 
This  added  piece  will 
then  extend  over  S{  in 
such  a  way  that  the  part 
of  the  plane  designated 
by  2n+i  is  doubly  cov 
ered  by  our  domain,  that 
is,  is  covered  by  two  "sheets."  We  think  of  these  sheets  as 
completely  separated  from  each  other  —  perhaps  by  supposing 
space  between  them.  The  domain  momentarily  defining  the  func 
tion  has  tJierefore  in  the  simplest  case  the  form  of  a  fiat  strip 
bounded  by  curved  lines,  the  ends  of  the  strips  extending  partly  one 
over  the  other  (Fig.  42). 

Of  course  one  case  then  the  other  can  appear  according  to 
the  direction  in  which  we  proceed  with  the  continuation.  But 
by  proceeding  with  each  new  continuation  in  the  prescribed 
manner  for  the  case  at  hand,  we  obtain  finally  the  entire  RIE 
MANN'S  surface  which  belongs  to  the  function  generated  by  the 
given  element  of  the  function.  We  say  : 

II.  The  totality  of  analytic  continuations  of  an  element  of  a  func 
tion  forms  in  general  a  many-valued  function  of  z,  which  however 


FIG. 


388       VI.  GENERAL  THEORY  OF  FUNCTIONS 

can  be  regarded  as  a  single-valued  function  of  position  on  a  suitably 
constructed  RlEMANN'S  surface. 

To  be  sure  it  is  possible  that  after  a  series  of  continuations 
which  have  led  to  different  values  of  the  function  for  the  same 
z  —  for  example  after  a  series  of  circuits  of  the  band  shown  in 
Fig.  42 — we  may  come  again  to  values,  or  more  exactly  to 
developments,  which  were  already  obtained  there.  In  this  case 
we  will  have  to  fuse  the  newly  generated  sheet  of  the  RIEM ANN'S 
surface  with  one  already  formed.  We  encounter  difficulties  here 
in  the  geometrical  representation  when  the  two  sheets  under 
consideration  do  not  lie  directly  over  each  other ;  we  must  then 
imagine  that  one  of  these  two  sheets  pierces  the  intermediate  ones 
at  bridges  (cuts)  in  order  to  be  combined  with  the  other.  But 
the  bridges  arising  in  this  way  are  not  essential  for  the  surface ; 
they  may  be  shifted  in  the  most  varied  way,  and  we  are  to 
keep  in  mind  in  this  connection  that  two  parts  of  the  surface 
crossing  in  a  cut  are  not  to  be  looked  upon  as  having  a  continu 
ous  connection  with  each  other.  We  were  acquainted  with  all 
these  details  in  treating  the  individual  functions  in  the  previous 
chapter  so  that  further  study  is  unnecessary  here.  Only  one 
possibility,  of  which  we  have  had  as  yet  no  example,  remains  to 
be  mentioned  :  Bridges  (cuts)  may  also  intersect  in  the  most 
varied  manner.  Of  course  we  seek  to  avoid  this  possibility 
when  it  occurs,  but  it  is  not  always  possible  to  do  so. 

We  may  also  think  of  the  RIEMANN'S  surface  as  spread  out 
over  the  sphere  instead  of  over  the  plane.  For  this  purpose  we 
map  the  neighborhood  of  the  point  at  infinity  upon  the  neighbor 
hood  of  the  origin  of  the  s'-plane  by  the  substitution : 

*'=i/* 

by  which  the  given  f  unction  /(z)  tranforms  into  a  function  ^(z'); 
we  then  study  this  function  <j>  (z1)  in  the  z'-plane.     If  the  origin 


§  68.    SINGULAR   POINTS  AND   NATURAL  BOUNDARIES      389 

of  the  z'-plane  can  be  reached  by  an  analytic  continuation  of  the 
function  <£  (2')  in  this  plane,  we  regard  the  point  oo  of  the 
2-sphere  as  belonging  to  the  domain  denning  the  function  f(z) 
upon  that  sphere. 

§68.     Singular  Points  and  Natural  Boundaries   of   Single-valued 

Functions 

When  the  analytic  continuations  of  an  element  of  a  function 
cover  the  whole  sphere  uniquely,  this  element  of  the  function 
generates  a  function  which  is  single-valued  over  the  whole 
sphere.  But  such  a  function  is  necessarily  a  constant  according 
to  IV,  §  44.  Hence  : 

I.  The  domain  for  which  a  single-valued  function  not  a  constant  is 
defined,  never  covers  the  entire  sphere. 

A  series  of  further  possibilities  thus  arise  for  discussion. 

The  case  is  at  once  conceivable  that  there  are  one  or  more 
points  which  lie  upon  the  boundary  of  the  domains  of  conver 
gence  of  certain  continuations,  but  which  do  not  lie  in  the  in 
terior  of  any  one  of  these  domains.  Let  us  consider  the  extreme 
case  where  we  have  only  one  such  point.  This  point  itself  can 
therefore  not  be  reached  by  the  continuations  of  the  function  but 
any  other  point  of  its  neighborhood  can  be  so  reached.  Thus 
the  definition : 

II.  A  point  such  that  it  cannot  be  reached  by  any  continuation 
of  the  function,  but  that  an\  otJier  point  of  its  neighborhood  can  be 
so  readied,  is  called  an  isolated  singular  point  of  the  function. 

The  behavior  of  a  function  in  the  neighborhood  of  such  a 
point  has  already  been  investigated  in  §§  43,  47.  48  ;  the  follow 
ing  is  a  recapitulation  of  that  investigation  : 

III.  An  isolated  singular  'point  of  a  single-valued  function  is 
either  a  pole  or  an  essential  singularity,  —  a  pole  being  a  point  at 


390       VI.  GENERAL  THEORY  OF  FUNCTIONS 

which  the  function  has  an  infinity  of  an  assignable  integral  order 
and  an  essential  singularity,  a  point  in  whose  neighborhood  the  func 
tion  approaches  arbitrarily  near  to  any  arbitrary  value  an  infinite 
number  of  times  * 

It  is  further  conceivable  that  the  function  has  an  infinite  num 
ber  of  poles.  The  totality  of  these  poles  considered  as  an  infi 
nite  set  of  points  must  necessarily  have,  therefore,  at  least  one 
limit  point  according  to  XVI,  §  25.  In  such  a  case  the  limit 
point  itself  cannot  belong  to  the  domain  for  which  the  function 
is  regular ;  for  then  the  function  would  have  to  be  regular  also 
in  a  neighborhood  of  this  point.  Moreover,  it  cannot  be  a 
pole ;  for,  according  to  IV,  §  43  a  circle  of  so  small  a  radius 
can  be  drawn  about  a  pole  such  that  no  other  singular  point  of 
the  function  lies  in  it.  Consequently,  we  say : 

IV.  We  designate  as  an  isolated  essential  singular  point  of  the 
function  such  a  point  in  whose  neighborhood,  arbitrarily  small,  infi 
nitely  many  poles,  but  no  other  singularity  of  the  function  lie,  provided 
that  this  point  is  isolated  not  from  poles,  but  from  other  essential 
singular  points  of  the  function  ;  and  it  is  classed  with  the  essential 
singular  points  of  Theorem  III  as  the  "  first  kind"  of  such  points . 

It  may  be  mentioned  without  proving  that  for  these  singular 
points  also,  the  theorem  holds  that  the  function  comes  infinitely 
often  arbitrarily  near  to  any  arbitrary  value  in  a  neighborhood 
as  small  as  we  please  about  one  of  these  points. 

V.  Further,  infinitely  many  essential  singular  points  of  the  first 
kind  may  "  accumulate  "  about  such  a  point  of  the  "  second  kind,'''1 
infinitely  many  such  points  of  the  "  second  kind"  about  such  a  point 
of  the  "third  kind,"  etc. 

These  possibilities  will  not  be  discussed  further. 

*  Cf.  Exs.  1-6  at  the  end  of  §  68.  —  S.  E.  R. 


§  68.    SINGULAR   POINTS   AND   NATURAL   BOUNDARIES      391 

However,  a  few  words  must  be  given  to  another  possibility, 
viz.  where  all  the  points  of  a  line  are  such  that  they  never  lie 
in  the  inside  of  the  domain  of  convergence  of  the  continuation 
of  a  given  element  of  the  function ;  and  we  are  to  understand 
the  word  line  here  in  the  most  general  sense  defined  in  IX, 
§  25.  Such  a  line  is  called,  therefore,  a  line  of  singularities  of 
the  function.  Its  points  may  lie  in  part  (to  be  sure  not  inside, 
but)  upon  the  boundary  of  the  domain  of  convergence  of  the 
analytic  continuation  of  the  original,  given  element  of  the  func 
tion  ;  but  the  case  can  also  arise  where  such  a  point  does  not 
lie  upon  the  boundary  of  such  a  domain  of  convergence.  By 
many  authors  only  the  points  of  the  first  kind,  not  the  points  of 
the  second  kind,  are  designated  as  singular  points  of  the  function. 

The  case  where  such  a  line  of  singularities  is  closed  is  of 
particular  interest.  It  delimits  then  a  region  of  the  surface 
beyond  which  the  function  cannot  be  continued ;  it  is  not  possi 
ble  on  the  basis  of  our  previous  agreements,  to  enlarge  the 
domain  for  which  the  function  is  defined  beyond  this  region  of 
the  surface ;  and  it  appears,  moreover,  that  such  enlargement 
of  the  domain  cannot  be  obtained  by  changing  or  supplement 
ing  these  stipulations.  On  the  contrary,  we  define : 

VI.  A  closed  line  of  singularities  of  a  function  is  a  natural 
boundary  for  the  function, 

Such  functions  with  natural  boundaries  do  not  appear  in  the 
elementary  parts  of  the  theory  of  functions,  but  examples  of 
such  functions  are  found  in  the  theory  of  elliptic  functions. 

Moreover,  these  natural  boundaries  of  analytic  functions  are 
always  to  be  distinguished  from  the  artificial  cuts  which  we  have 
used  at  times  to  separate  the  totality  of  values  of  a  many-valued 
function  into  distinct  branches  for  the  purpose  of  better  studying 
them ;  beyond  such  a  cut  the  analytic  continuation  takes  place 
in  another  branch. 


392  VI.    GENERAL  THEORY   OF   FUNCTIONS 

EXAMPLES 

1.  The  essential  singularity  may  be  contrasted  as  follows: 
If  the   reciprocal  of  the  function  has  a  point  for  an  ordinary 
point,   this  point  is  a  pole,  that  is,  it  is,  to  be  sure,  a  zero  for 
the  reciprocal  of  the  function  ;  but  when  the  value  of  the  recip 
rocal    of  the  function    is   not   determinate    at  the    point,  then 
the  point  is  an  essential  singularity  for  the  function  as  well  as 
the  reciprocal. 

2.  Consider   the  function  e^/z.     Show  that  as  z  approaches 
zero,  this  function,  elsewhere  one-valued,  may  be  made  to  ap 
proach  any  arbitrary  value,  that  is,  z=  o  is  an  "essential  singu 

larity."  2 

HINT:  e*  =  I  +z+  —+   ••• 
2  ! 


Therefore,  z1  =  o  or  z  =  oo  is  an  essential  singular  point,  that  is, 
there  is  no  number  m  such  that  zm  times  a  power  series  in  z  is 
holomorphic,  that  is,  z'  =  o  gives  an  infinite  number  of  infinities. 
Thus  z'  =  o,  z  =  oo  is  an  essential  singularity. 

3.  Show  by  using  Ex.  2  that  in  the  vicinity  of  an  essential 
singular  point  an  infinite  number  of  poles  exist. 

4.  Discuss  e^a  for  its  poles  and  essential  singular  point. 

5.  Discuss  sin  z,  i/sin  z,  i/sin  (1/2)  as  in  Ex.  4. 

6.  Rational  functions  have  a  finite  number  of  poles  ;   tran 
scendental   functions  are  everywhere  holomorphic  except  they 
have  at  least  one  essential  singular    point.     Rational  integral 
functions  and  transcendental  integral  functions  are  holomorphic 
everywhere  in  the  finite  part  of  the  plane,  but  one  has  poles  at 
infinity  while  the  other  has  an  essential  singularity  at  infinity. 
Give  illustrations. 


§  69.    SINGULAR   POINTS   AND   NATURAL   BOUNDARIES      393 

7.    If  /j.  (z)  and/2(X)  are  any  two  one-valued  analytic  functions 
of  z  with  a  finite  number  of  singular  points,  then  the  expression 


, 


defines,  inside  and  out  of  the  unit  circle  about  the  origin,  parts 
of  two  distinct  analytic  functions,  /i  (z),  f.2  (z).  Show  that  the 
circle  itself  is  not  a  natural  boundary  for  either  of  these 
functions. 

§  69.     Singular  Points  and  Natural  Boundaries  of  Many-valued 
Functions 

If  we  are  studying  a  many-valued  function,  then  considera 
tions  analogous  to  those  carried  out  in  the  previous  paragraph 
for  the  plane  are  to  be  made  for  the  RIEMANN'S  surface  upon 
which  the  many-valued  functions  to  be  investigated  is  a  single- 
valued  function  of  position.  We  must  then  speak  of  singular 
points  and  lines  in  a  distinct  sheet  ;  it  is  not  at  all  necessary 
that  such  points  and  lines  which  appear  in  the  different  sheets 
be  situated  vertically  over  each  other.  In  particular  it  is  not 
necessary  that  all  parts  of  the  2-plane  be  covered  by  the  same 
number  of  sheets  of  the  surface. 

But  many-valued  functions  have  other  singular  points  of  a 
different  kind,  viz.,  the  branch-points.-  We  have  already  had 
a  number  of  examples  of  such  singularities  in  the  previous  chap 
ter  ;  according  to  present  considerations  we  obtain  them  in 
general  as  follows  :  Let  a  point  a  be  given  and  a  circle  about  it 
as  center  with  a  sufficiently  small  radius  ;  let  b  be  a  point  inside 
of  this  circle  and  different  from  a.  Let  an  element  of  the  func 
tion  be  given  about  b  ;  we  limit  the  discussion  to  such  continu 
ations  of  this  element  which  can  be  obtained  without  going 
outside  of  this  circle.  It  is  then  possible  that  none  of  these  con- 


394       VI-  GENERAL  THEORY  OF  FUNCTIONS 

tinuations  reach  the  point  a,  that  they  reach  every  other  point 
inside  of  the  circle,  but  that  continuation  along  a  smaller  circle  con 
centric  to  the  first  only  leads  to  the  original  element  after  n  circuits 
(n  >  i).  In  this  case  n  sheets  of  our  RIEMANN'S  surface  are 
connected  at  a  exactly  as  is  exhibited  in  §  63  in  the  investigation 
of  the  function 


(1)  w  =  vs  —  a 

(studied  for  a  =  o).  If  we  map  the  parts  of  the  n  sheets  lying 
inside  of  the  first  named  circle  upon  the  #/-plane  by  means  of 
this  function,  then  the  images  of  these  sheets  are  arranged 
smoothly  and  contiguously  in  this  plane  and  cover  the  neighbor 
hood  of  the  origin  uniquely.  The  function  f(z)  to  be  investi 
gated  is  thus  transformed  into  a  function  of  w,  $(w),  whose 
particular  branch  under  consideration  is  regular  at  each  point 
of  the  neighborhood  of  the  origin,  excepting  the  origin  itself, 
and  which  returns  into  itself  after  one  circuit  about  the  origin. 
If  we  can  now  show  that  the  value  of  f(z)  remains  less  than 
an  assignable  limit  however  near  z  may  approach  a  in  any 
direction,  then  <£(w/)  also  remains  less  than  this  limit  when  w 
approaches  the  origin  arbitrarily.  But  then  the  origin  cannot 
be  a  singular  point  of  <f>(w)  according  to  I,  §  48  ;  on  the  con 
trary  <fr(w)  is  regular  at  the  origin,  and  can  be  developed  in  a 
MACLAURIN'S  series.  Expressing  w  in  this  series  in  terms  of  z 

we  obtain : 

Li  ™ 

(2)  /(*)=  a,  +  ai(z  -  a)n  +  a,(z  -  a)n  +  -  +  am(z  -  a)n  +  »•. 

Here,  as  follows  from  the  derivation,  any  one  of  the  values  of 

this  ;/-valued  function  can  be  chosen  for  (z  —  a)n ;  the  values  of 
the  remaining  terms  of  the  series  are  then  no  longer  arbitrary, 

H 

since  in  general  we  are  to  take  for  (z  —  a)H  the  mih  power  of 
the  value  chosen  for  (z  —  a)n.     For  every  value  of  z  considered, 


§  69.    SINGULAR   POINTS   AND   NATURAL  BOUNDARIES      395 

the  series  (2)  therefore  represents  ;/  values  of  the  function  in 

± 

accordance  with  the  ;/  values  of  (z  —  af\  together  they  consti 
tute  the  n  branches  of  the  function  f(z)  which  are  connected 
cyclically  about  a.  Such  a  point  is  called  a  branch-point  or 
winding-point  of  order  («—  i);  we  assign  it  to  the  domain  in 
which  we  have  defined  the  function,  and  ascribe  to  the  function  at 
this  point  tlie  value  <70. 

But  if  we  cannot  show  that  f(z]  and  <fxw']  remain  less  than 
a  finite  limit  in  the  neighborhood  of  z  —  a  and  w  =  o  respec 
tively,  we  cannot  apply  MACLAURIN'S  theorem  for  the  develop 
ment  of  <f»(ni) ;  but  we  can  use  LAURENT'S  theorem  for  this  pur 
pose.  In  this  way  f(z)  is  developed  in  a  series  of  powers  of 
z  —  a  whose  exponents  are  positive  and  negative  fractions  with 
n  as  denominator.  Such  a  point  is  said  to  be  a  branch-print  and 
a  singular  point  at  the  same  time;  it  is,  in  fact,  a  pole  or  an 
essential  singular  point  according  as  the  development  just  men 
tioned  contains  a  finite  or  an  infinite  number  of  terms  with 
negative  exponents. 

We  may  also  have  branch-points  at  which  infinitely  many 
sheets  are  joined  together ;  we  have  had  an  example  of  this  in 
studying  the  logarithm.  But  we  shall  not  enter  here  into 
further  discussion  of  such  points,  as  also  points  in  whose  neigh 
borhood  infinitely  many  branch-points  are  accumulated. 

We  now  take  up  a  question  postponed  in  §  34,  viz.,  the  ques 
tion  as  to  the  conformality  of  the  representation  determined  by 
a  regular  function  in  the  neighborhood  of  a  point  at  which 
dw/dz  —  o.  Without  loss  of  generality  we  may  suppose  the 
point  we  are  considering  to  be  the  origin  of  the  s-plane  and 
that  the  origin  of  the  a'-plane  corresponds  to  it ;  let  the  develop 
ment  of  w  in  powers  of  z  have  the  form : 

(3)  ™  =  a*zn  +  0»+i3n+1  + 


396  VI.    GENERAL  THEORY   OF   FUNCTIONS 

and  let  #0  be  different  from  zero.     If   we    then    introduce    an 
auxiliary  variable  s  by  the  equation  : 

(4)  w  =  sn, 
we  obtain 

(5)  j 


The  principal  value  of  the  ;/th  root  of  the  quantity  in  paren 
thesis  is  regular  in  the  neighborhood  of  the  origin  (cf.  I,  §  61)  ; 
hence  in  the  neigborhood  of  z  —  o,  s  is  a  regular  function  of  z 
whose  derivative  for  z  =  o  is  not  zero  but  is  equal  to  -\/an.  The 
relation  between  the  .r-plane  and  the  s-plane  is  therefore  con- 
formal  at  the  origin  ;  but  on  account  of  equation  (4)  and  §  18 
the  angle  at  the  origin  in  the  w-plane  is  n  times  as  large  as  the 
angle  at  the  origin  in  the  .r-plane.  And  therefore  the  angle  at 
the  origin  in  the  w-plane  is  n  times  as  large  as  the  correspond 
ing  angle  at  the  origin  in  the  s-plane  ;  in  other  words  we  have 
the  theorem  : 

If  a  function  is  regular  at  a  point  in  the  z-plane  and  if  the  first 
(n—\]st  derivatives  of  this  function  are  equal  to  zero  at  this 
point,  but  the  nth  derivative  is  different  from  zero,  then  in  the  trans 
formation  from  this  plane  to  the  w-plane,  the  angle  at  this  point 
increases  n-fold. 

According  to  X,  §  46,  z  is  then  a  regular  function  of  s  =  wl/n 
in  the  neighborhood  of  s  —  o  in  whose  development  the  coeffi 
cient  of  the  first  term  is  not  equal  to  zero  ;  thus  w  =  o  is  a 
branch-point  of  order  n  —  i  for  the  inverse  function  z(w).  Since 
these  considerations  are  reversible,  it  follows  that  : 

If  the  development  of  a  function  z(w]  in  the  neighborhood  of  an 
(n  —  \\fold  branch-point  a,  begins  with  (w  —  a)l/n  following  a  con 
stant  term  #0,  then  the  angle  at  this  point  is  reduced  to  its  nth  part 
in  the  transformation  from  the  w-plane  to  the  z  plane. 


§  70.   ANALYTIC  FUNCTIONS  OF  ANALYTIC  FUNCTIONS      397 

To  a  line  of  the  a'-plane  which  has  a  definite  tangent  at  the 
point  w  =  «,  corresponds  then  a  line  of  the  s-plane,  which  has 
the  point  z  —  ac  as  an  #-fold  point  with  ;/  separate  tangents ; 
these  tangents  form  angles  ir/n  with  each  other. 

EXAMPLES 

1.  State  the  theorem  concerning  isolated  singular  points  of 
analytic  functions  at  which  the  function  remains  finite. 

2.  Assuming  the    theorem    in    Ex.    i,    establish    other    facts 
about  isolated  singular  points,  and  deduce  the  form  of  develop 
ment  of  a  function  about  a  pole. 

3.  Given  the  function  f(z}=^n\  show  that  the  circle  of 

n=l 

convergence,  that  is,  the  unit  circle  about  the  origin,  is  a  natural 

boundary. 

lBd 

HINT.  —  If  q  and  r  are  integers,  the  point  e  r    on  the  circle  of  conver- 

a^i 

gence  is  an  obstacle  to  the  continuation.  For,  put  z  =  p  •  e  r  (p  <  i)  and 
let  p  increase;  then  as  p  approaches  I,  the  part  of  the  series  from  the  rth 

term  onwards,  namely  ^  p"!  approaches  infinity.     This  would  be  impossible 

2qTTi  n=r 

if  the  point  e  r  were  situated  inside  of  any  immediate  continuation  of  the 
power  series.  It  is  thus  clear  that  there  are  infinitely  many  obstacles  on  the 
circle  of  convergence  and  too  that  on  any  arc  there  are  infinitely  many 
obstacles. 

§  70.  Analytic  Functions  of  Analytic  Functions 
If  z'  is  an  analytic  function  of  z: 

(i)  *'  =  <K*) 

and  if  w  is  an  analytic  function  of  z  : 

(2)  w  =/*'), 
the  question  arises  whether 

(3)  w  =  F(s) 


3Q8       VI.  GENERAL  THEORY  OF  FUNCTIONS 

is  an  analytic  function  of  z  and  in  what  sense  it  is  such  a 
function. 

We  have  already  disposed  of  the  simplest  case  in  X,  §  38. 
If  (f)  is  single-valued  and  regular  in  a  domain  S  of  the  z-plane, 
and  if  all  the  values  of  <£  which  belong  to  points  of  this  domain 
fall  in  a  domain  S'  of  the  z'-plane  in  which  domain/  is  regular, 
then  w  is  also  regular  in  S. 

But  if  <£  or  /or  both  are  many-valued  functions,  the  question 
arises :  When  we  give  all  their  values  to  these  two  functions  in 
(3),  will  the  totality  of  values  of  w  so  obtained  belong  to  one 
and  the  same  analytic  function  of  z,  or  to  different  functions  of 
this  kind  ?  and  in  both  cases :  will  this  function  (or  these  func 
tions)  be  obtained  completely  in  this  way,  or  are  there  still  other 
values  belonging  to  it  (or  to  them)  ?  To  answer  this  question 
we  must  follow  the  analytic  continuation  somewhat  in  detail ; 
and  according  to  XII,  §  54,  it  will  be  sufficient  to  limit  ourselves 
to  closed  paths  in  doing  so. 

Let  then  ZQ  be  a  value  of  z  for  which  the  function  <£  takes  on 
a  value  z'Q  (along  with  other  values).  Let  the  function  /(/) 
be  defined  for  z\  and  let  its  value  or  one  of  its  values  be  w0. 
Let  the  corresponding  elements  of  the  function  be  denoted  by 
(z'o),  (w0).  The  function-symbol  f(z')  includes  then  besides  w0, 
all  the  values  which  are  obtained  by  allowing  z'  to  take  on  all  the 
values  on  arbitrary  closed  paths  in  its  plane  and  continuing  w, 
beginning  with  WQ,  analytically  as  a  function  of  z' ;  on  the  con 
trary  the  function-symbol  F(z)  includes  the  values  which  are 
obtained  by  allowing  z  to  take  on  the  values  on  closed  paths  in  its 
plane  and  in  this  way  continuing  w  analytically  as  a  function  of 
z.  The  question  as  to  whether  the  two  functions  are  identical 
or  different  is  thus  reduced  to  the  two  following  cases : 

I.  Can  z'  be  continued  along  all  the  s-paths  upon  which  F(z) 
can  be  continued?  This  is  then  and  only  then  not  the  case 


§  7i.    THE   PRINCIPLE   OF   REFLECTION  399 

when  <f>  (z)  has  natural  boundaries  which  are  not  such  boundaries 
for  F(z).*  The  function-symbol  F(z)  has  then  a  wider  mean 
ing  than/[(£(s)]. 

II.  Can  any  closed  z'-path  be  obtained  by  allowing  z  to 
describe  a  suitable,  closed  path  in  its  plane?  This  is  then 
not  the  case  when  z  =  *J/(z'),  the  inverse  of  the  function 
z'  =  <£(s),  is  not  single-valued.  In  this  case  the  analytic  func 
tion  F(z)  can  include  perhaps  f  only  a  part  of  the  values  of 
f\_4>(z)~\.  We  have  had  an  example  of  this  in  logs2  in  §  56; 
w^/znp  is  a  second  example. 

The  remaining  values  of  /[<XS)]  are  classified  as  other  ana 
lytic  functions  F^z),  Fz(z),  •••  so  that  /[>(>)]  is  thus  divided 
into  a  (finite  or  an  infinite)  number  of  such  functions. 

(Cases  I  and  II  may  both  apply  to  the  same  function  ;  then 
only  a  part  of  the  values  of  /[<£(z)]  are  identical  with  a  part  of 
the  values  F(z).) 

The  relations  become  more  complicated  if  we  assume  f  to 
depend  not  upon  one  but  upon  two  (or  more)  functions  of  z, 
<£(z),  x(z)-  But  the  discussion  of  functions  of  several  complex 
variables  is  excluded  from  this  book ;  let  it  be  said,  however, 
that  in  this  case  <£,  x  are  to  be  continued  simultaneously  in  order 
to  obtain  values  of  F(z)  and  thus  we  are  not  always  free  to 
associate  two  arbitrary  values  of  <f>  and  x- 

§  71.     The  Principle  of  Reflection 

The  general  method  of  analytic  continuation  developed  in 
§  67  is  not  suited  for  actual  application  in  investigating  particu 
lar  functions.  It  is  best  in  such  cases  to  resort  to  special 

*  That  this  can  happen  is  shown  by  the  trivial  example  that  /  is  the  inverse  of 
4>  and  thus  F(z)  =  z. 

f  This  is  not  necessarily  the  case ;  the  values  of  F  considered  can  perhaps  be 
obtained  when  z  describes  other  paths  ;  ^/**t  where  m,  n  are  prime,  is  an  example 
of  this. 


40O       VI.  GENERAL  THEORY  OF  FUNCTIONS 

methods :  an  important  method  of  this  kind  is  discussed  in  this 
and  the  following  paragraphs. 

Let  us  fix  in  mind  a  very  special  case.  Let  a  f unction  f(z)  be 
by  definition  regular  in  the  interior  of  a  domain  A  of  the  s-plane, 
a  part  of  whose  boundary  is  a  piece  of  the  axis  of  real  numbers. 
If  the  function  is  also  regular  and  real  on  this  piece  of  the  axis 
and  if  ZQ  is  a  point  on  this  piece,  its  development  in  powers  of 


FIG.  43 

z  —  ZQ  has  real  coefficients,  and  therefore  it  takes  on  conjugate 
complex  values  at  pairs  of  conjugate  complex  s-points.  But  we 
will  not  assume  from  the  start  that  the  function  is  regular  on  this 
piece  of  the  axis  nor  even  on  only  a  part  of  this  piece  ;  we  sup 
pose  only  that  as  z  approaches  arbitrarily  to  a  definite  point  x  of 
this  piece  of  the  axis,/(Y)  converges  to  a  definite  real  value  f(x) 
in  the  limit,  and  that  these  values  of  f(x)  together  with  the 
values  of  the  function  /(z)  at  interior  points  of  A  form  a  continu 
ous  function  of  the  real  variables  x  and  y.  * 

Let  us  now  determine  the  points  z  conjugate  to  the  points  z 

*  We  do  not  discuss  here  whether  this  second  supposition  is  a  consequence  of 
the  first:  concerning  this  see  P.  PAINLEVE,  Ann.de  lafac.  de  Toulouse,  Vol.  II 
(1888),  p.  19. 


§  7i.    THE   PRINCIPLE  OF   REFLECTION  401 

of  the  domain  A  ;  all  these  ^-points  thus  form  a  domain  A  which 
is  the  reflection  of  A  in  reference  to  the  axis  of  real  numbers. 
Then  by  assigning  to  each  point  z  that  value  which  is  conjugate 
to  the  value  oif(z)  at  z,  we  define  a  function  which  is  regular  in  A> 
viz. : 

(i)  /.<=)=/«. 

Let  £  be  a  point  in  the  interior  of  A  :  it  then  follows  according 
to  the  theorem  of  CAUCHY  *  that : 


(2)  sbJui-*  *"/<0 

but  that : 


We  now  add  these  two  equations  member  by  member.  The 
parts  of  the  two  integrals  taken  along  the  piece  of  the  axis  of 
reals  thus  drop  out  since  z  =  z,/(z)  =fi(z)  along  this  piece  of 
the  axis,  and  the  direction  of  integration  is  in  the  one  case 
opposite  to  that  in  the  other  ;  /(£)  remains  ;  it  is  expressed  by 
an  integral  taken  along  the  boundary  of  the  domain  (A  -\-  A), 


where  ^  -  =  z,  f-2(f)  =f(z)  along  the  part  of  the  boundary  belong 
ing  to  A  and  ^  =  z,/z(f)  =/i  (z)  along  that  part  belonging  to  A.  f 
This  integral  has  the  exact  form  of  the  CAUCHY'S  integral  ;  but 
such  an  integral  represents  a  function  regular  in  the  whole 
domain  and  designated  here  temporarily  by  <£(£)  (cf.  cor.  to  I, 

*  In  order  to  apply  CAUCHY'S  theorem  here  we  must  apply  it  to  a  curve  which 
lies  entirely  inside  of  A,  and  then  pass  from  this  to  the  boundary  of  the  domain 
with  the  aid  of  III,  §  29. 

f  The  symbol  for  the  variable  of  integration  can  be  selected  arbitrarily. 


402  VI.    GENERAL  THEORY   OF  FUNCTIONS 

§  36,  also  the  first  proof  of  I,  §  50).     The  process  thus  shows 
that/(£)  =  <£(£)  in  the  domain  A. 
But  if  £  is  a  point  in  A  we  have 

(4) 


and 

(5) 


Proceeding  as  in  the  case  above  we  find  that  this  same  regular 
function  </>(£)  is  identical  with  fi(£)  in  the  domain  (A).  There 
is  therefore  a  function  regular  in  the  whole  domain  (A  -f-  A), 
which  is  identical  with/(£)  inside  of  (A)  and  identical  with/i(£) 
inside  of  A  ;  but  this  means  precisely  that  f\(Q  is  the  analytic 
continuation  of  /"(£)•  Hence  the  theorem  : 

I.  The  analytic  continuation  of  f(z)  across  this  piece  of  the  real 
axis   is  always  possible  under  the  given  assumptions  ;   it  is  per 
formed  by  assigning  conjugate  values  of  the  function  to  conjugate 
values  of  the  argument* 

The  theorem  may  be  easily  generalized  to  the  case  where 
any  other  straight  line  of  the  plane  is  used  instead  of  the 
axis  ;  hence  : 

II.  If  an  analytic  function  takes  on  real  values  (in  the  sense  de 
fined  at  the  beginning  of  the  paragraph)  along  a  piece  of  a  straight 
line,  it  takes  on  conjugate  complex  values  at  such  points  which  are 
reflections  of  each  other  in  reference  to  that  line. 

*  This  particular  continuation,  important  in  investigations  concerning  con- 
formal  representation,  is  contained  in  a  proposition  due  to  SCHWARZ,  Crelle,  Vol. 
70  (1869),  pp.  ic6,  107,  Ges.  Math.  Abh.  Vol.  II,  pp.  66-68.  Cf.  also  DARBOUX, 
Theorie  generate  des  surfaces,  Vol.  i,  §  130.  —  S.  E.  R. 


72.   CONFORM AL  REPRESENTATION  OF  A  TRIANGLE      403 


§  72.    Conformal  Representation  of  a  Triangle  bounded  by 
Straight  Lines  on  a  Half-plane 

The  theorems  of  the  previous  paragraph  may  be  applied  to 
the  solution  of  the  following  problem  :  to  map  a  triangle 
bounded  by  straight  lines  in  the  o/-plane  conformally  on  a  z- 
half-plane  (or  on  a  s-hemisphere).  Let  us  suppose  the  possibil 
ity  of  the  solution  aad  designate  by 

(i)  z  =  <tfw) 

the  function  desired  for  the  mapping.     Primarily  the  problem 

implies  that  this  function  be  regular  inside  of  the  triangle,  that  it 

remains  continuous  in  approaching  the  sides  of  the  triangle,  and 

that  it  assumes  real  values  on  them.     But  these  properties  are 

sufficient   according  to  II, 

§  71  to  continue  the  func 

tion  analytically  beyond  the 

sides   of  the  triangle  over 

the    three    other   triangles, 

which  are  the  reflections  of 

the  given  triangle  in  refer 

ence    to     its     sides.     The 

same  conclusion   can   then 

be  applied  to  each  side  of 

each  of  the  new  triangles, 

etc.,  so  that  finally  the  whole 

RIEMANN'S  surface  of  <£(#/) 

is  constructed  entirely  from 

triangles  which  are   alternately  congruent  and   symmetrical  to 

the  given  one.     In   this  way  the  triangles  formed  later  over 

lap  those   formed   earlier,  even  the  original  triangle,  and    the 

RIEMANN'S  surface  so  formed  will  in  general  be  composed  of 

an  infinite  number  of  sheets.     In  order  that  the  surface  be  one- 


FlGl 


404 


VI.    GENERAL  THEORY   OF   FUNCTIONS 


sheeted  it  is  necessary  that  one  vertex  of  the  triangle  shall  not 
be  a  branch-point,  and  that  therefore  we  shall  again  obtain  the 
original  triangle  after  an  even  number  of  reflections  on  the  sides 
of  the  triangle  intersecting  at  such  a  vertex.  For  this  purpose 
it  is  necessary  and  sufficient  that  each  angle  of  the  triangle  be  an 
aliquot  part  of  IT. 

But  when  this  condition  is  satisfied  the  plane  is  always  cov 
ered  uniquely  by  the  alternately  congruent  and  symmetrical 
repetitions  of  the  original  triangle.  This  is  best  illustrated  by 
examining  the  possible,  individual  cases  of  which  there  are  only 
a  small  number.  For,  if  the  angles  of  a  triangle  are  TT//,  TT/;;Z, 
Tr/n,  where  /,  m,  n  are  integers  >  i,  these  integers  must  satisfy 
the  equation 
(2)  i//+i/»i  +  i/«=i. 

This  is  at  once  the  case  when  each  =  3  and  the  triangle  is 
therefore  equilateral.  But  if  they  are  not  all  equal  to  3,  one 
must  be  smaller  and  hence  =  2.  Let  /=  2  ;  i/m  +  i/n  is  then 

=  1/2,  and  thus  (m  —  2)  (n  —  2)  =  4, 
and  therefore  either  m  =  4,  n  =  4,  or 
;«  =  3,  n=6  (m  =  6,  11=3  is  the  same 
case).  Therefore : 

I.  The  surface  of  a  triangle  bounded 
by  straight  lines  can  be  mapped  conform- 
ally  upon  a  ha  If  plane  in  only  three 
cases  by  means  of  a  function  which  is 
single-valued  in  the  whole  plane,  viz.  : 
when  the  triangle  is  cither  equilateral, 
or  right-angled  isosceles,  or  half  of  an 
equilateral  triangle. 

We  notice  now  that  a  parallelogram 
FIG.  45  is  formed  from  eighteen  of  these  alter- 


§  72.   COXFORMAL  REPRESEXTATIOX  OF  A  TRIAXGLE      405 


nately  congruent  and  symmetrical  triangles  in  the  first  case,  from 
eight  of  them  in  the  second  case,  and  from  twelve  in  the  third 
case  ;  this  parallelogram  is  such  that  further  continuation  always 
leads  *  to  congruent  f  parallelo 
grams.  That  the  plane  is  cov 
ered  once  without  gaps  by 
congruent  parallelograms  is  only 
an  elementary  theorem. 

The  functions  which  determine 
the  representation  can  be  ob 
tained  as  follows  in  every  case 
(not  merely  for  the  three  special 
cases  mentioned  above)  : 

Let  w  =f(z)  be  the  solution  of 


Fic" 


equation  (i)  :  then  the  half-plane  is 
mapped  on  a  triangle  similar  to  the 
given  one  by  the  function  C^IL<  +  C2 
where  C^  C2  are  arbitrary  constants 
(cf.  §  10).  The  indefiniteness  arising 
in  this  way  is  eliminated  by  consider 
ing  the  function 


(3) 


d_ 

dz 


dw 
~dz 


FIG.  47 


instead  of  the  function  w ;  this  func 
tion  (3)  remains  unchanged  when 
C^w  +  C2  is  substituted  for  w.  With 
out  loss  of  generality  we  may  further 


*  The  number  eighteen  of  the  first  case  can  be  reduced  to  six  by  constructing 
the  parallelogram  from  parts  of  different  triangles.  This  is  designated  in  Fig.  45 
by  the  dotted  lines. 

t  Congruent  also  in  reference  to  the  position  of  the  individual  triangles  in  them 
which  are  indicated  in  the  figures  by  hatching. 


406  VI.    GENERAL  THEORY   OF   FUNCTIONS 

suppose  that  the  three  points  o,  i,  oo  of  the  sphere  taken  in 
order  correspond  to  the  three  vertices  of  the  triangle  ;  for, 
according  to  §  15  this  can  always  be  done  previously  by  a  linear 
transformation  of  the  z  variables.  Then  w  must  be  a  function 
of  z  which  is  regular  in  the  neighborhood  of  every  point  of  the 
^-sphere  with  the  exception  of  the  three  points  just  named  and 
which  has  a  derivative  different  from  zero  (VI,  §  34).  The 
angle  TT  of  the  z  half-plane  must  be  mapped  upon  the  angle  «?r 
of  the  triangle  of  the  w-plane  at  the  point  z  =  o;  hence  at  this 
point  we  must  have 

W    —    W      =    Z°-,'z 


W  a-l        /•  /    \ 

_  =  «.,    >./,(«), 


(4) 


where  /(z),  fi(z),  fi(z)  are  understood  to  be  functions  regular  in 
the  neighborhood  of  the  origin.  Similarly,  in  the  neighbor 
hood  of  the  point  i  : 

(15)  —  log  —  —  ^  ~  l  +  a  regular  function  ; 

dz        dz       z  —  i 

and  in  the  neighborhood  of  the  point  oo  : 

(6)  -^  log  ~  =  ~HH-!  +  z-*  •  a  regular  function. 

dz        dz  z 

Therefore  the  function  (3)  has  poles  of  the  first  order  at  the 
singular  points  o  and  i,  it  is  otherwise  regular  over  the  whole 
sphere,  and  is  zero  at  infinity  ;  it  is  consequently  a  rational 
function  according  to  VI,  §  44,  and  is,  in  fact, 


§  73-   GENERALIZATION  OF  PRINCIPLE  OF  REFLECTION      407 

(The  development  of  this  function  in  the  neighborhood  of  z=^c 
takes  the  form  (6)  since  a  -f-  (3  +  y  =  i.)  Integrating  (7)  twice 
we  obtain  : 


as  the  solution  of  the  problem  ;  further  discussion  of  this  solu 
tion  is  beyond  the  bounds  set  for  our  purpose. 

The  limiting  case  «  =4-,  /?  =  i  (and  thus  /=  2,  m  =  2,  n  =  oo) 
leads,  if  we  put  2  s  =  £  H~  i,  to  the  mapping  of  a  half-strip  on 
the  half-plane  by  the  function  w=-  sin-1£  investigated  in  §§  42 
and  62  d. 

§  73.   Generalization  of  the  Principle  of  Reflection  ;  Reflection  on 

a  Circle 

The  theorem  of  §  7  1  is  capable  of  a  very  wide  generalization 
as  worked  out  by  H.  A.  SCHWARZ.  Let  the  two  equations 

(1)  *  =  *(/),   }'=+($, 

in  which  <£,  \f/  signify  -at  present  real  regular  functions  of  the 
real  variable  /  (limited  to  a  definite  interval),  determine  a 
"  regular  arc  of  a  curve  "  ;  we  can  then,  according  to  I,  §  38, 
give  complex  values  to  this  variable  /  without  affecting  the  con- 
vergency  of  the  series  for  <£  and  \f/.  Therefore  by  the  equation 

(2)  s  =  ..r  +  /v=<K/)+/V<0» 

a  domain  of  the  /-plane  which  lies  on  both  sides  of  a  definite 
piece  of  the  real  /-axis,  is  mapped  on  a  domain  of  the  z-plane 
which  lies  on  both  sides  of  the  given  regular  arc  of  a  curve  ; 
and  we  can  restrict  the  first  domain  in  such  a  way  that  the 
latter  one  does  not  overlap  itself,  (X,  §  46).  If  the  s-points  are 
now  arranged  in  pairs  corresponding  to  conjugate  values  of  /  by 
means  of  (2),  we  define  in  this  way  in  the  last  named  domain 
a  reversibly  unique  arrangement  of  the  points  z  in  pairs. 


408       VI.  GENERAL  THEORY  OF  FUNCTIONS 

I.  This  arrangement  is  only  dependent  tipon  the  given  arc  of  a 
curve  itself,  and  independent  of  the  way  it  is  represented  by  equa 
tions  of  the  form  (/). 

In  order  to  obtain  another  way  of  representing  the  same  arc 
of  a  curve,  we  replace  /  in  equation  (i)  by  a  real,  regular  func 
tion  of  another  real  variable  T,  and  then  give  to  T  conaplex 
values  also  ;  in  this  way  conjugate  complex  values  of  /  corre 
spond  to  conjugate  complex  values  of  T  according  to  §  71. 
Accordingly  we  define  as  follows  : 

II.  Two  points  of  the   z-plane  which   correspond  to  conjugate 
points  of  the  t-plane  are  called  reflected  images  of  each  other  in 
reference  to  the  given  regular  arc  of  a  curve. 

Hence  the  following  more  general  theorem  is  obtained  from 
the  special  one  I,  of  §  71  : 

III.  Let  f(z]  be  a  function    regular  by  definition  inside  of  a 
domain  of  the  z-plane,  to  whose  boundary  a  regular  arc  of  a  curve 


belongs  ;  let  it  be  further  known  thatf(z]  converges  to  a  definite  real 
value  x(/)  in  the  limit  as  z  approaches  arbitrarily  to  a  definite  point 
t  of  this  arc,  and  that  these  limits  together  with  the  given  values  of 
the  function  forjn  a  continuous  function  of  x  and  y.  Then  the 
function  f(z]  may  be  continued  analytically  beyond  that  arc  of  the 
curve,  and  in  so  doing  it  takes  on  conjugate  complex  values  at  points 
which  are  reflected  images  of  each  other  in  reference  to  that  arc. 

If  in  particular  the  given  arc  is  an  arc  of  the  unit  circle,  we 
can  put 


and  hence  (cf.  3,  §  15)  : 


§  74-    MAP  OF  SPHERICAL  TRIANGLE  ON  HALF-PLANE      409 

If  we  now  give  to  t  in  this  equation  two  conjugate  values  //  +  ir 
and  u  —  iv  and  designate  the  corresponding  values  of  x  -f-  iy  by 
Xl  _j_  iy^  and  x2  +  iy2  respectively,  we  obtain  : 


i  -f  ?'  —  l 
i  +  v  +  // 


J      I     ,-,  ,°1/  I 

and  therefore  :  xz  —  n'2  =  —         -—  =  —   r~r~« 

i  —  z;  +  ///      A'!  +  ty\ 

But  that  is  exactly  the  relation  between  the  two  points  (x^  y^) 
and  (xz,  jo),  which  we  designated  earlier  as  reflection  on  the 
unit  circle  (cf.  equation  (7),  §  IT)  ;  hence  we  say: 

IV.  The  reflection  on  the  unit  circle  investigated  earlier  is  a 
special  case  of  the  reflection  on  an  arbitrary  regular  arc  of  a  curve 
defined  by  III. 

§  74.    Conformal  Representation  of  a  Triangle  bounded  by  Arcs  of 
Circles  upon  the  Half-plane 

In  §  72  we  made  use  of  the  special  theorem  of  §  71  to  investi 
gate  the  conformal  mapping  of  a  triangle  bounded  by  straight 
lines  upon  the  half-plane ;  the  more  general  theorem  of  §  73  is 
now  used  to  discuss  the  same  problem  for  a  triangle  bounded 
by  arcs  of  circles.  However,  the  present  problem  is  treated 
less  exhaustively  than  the  other  one  ;  we  limit  the  discussion  to 
emphasizing  a  few  particular  points  and  solving  an  easy  example. 

If  the  converse  of  the  function  used  for  the  mapping  is  to  be 
single-valued,  the  angles  of  the  triangle  must  be  aliquot  parts 
of  TT  in  this  case  also.  But  the  relation  (2),  §  72  is  not  neces 
sarily  satisfied  here ;  we  have,  consequently,  three  cases  to 
discuss,  viz. ; 


410       VI.  GENERAL  THEORY  OF  FUNCTIONS 

I.  If  i//  +  i/m  +  i/n  =  i,  we  show  geometrically  (cf.  Fig.  48) 
that  the  three  circles  to  which  the  arcs  bounding  the  triangle 
belong  intersect  in  a  point.  If  by  means  of  a  linear  transforma 
tion  we  pass  from  the  w-plane 
to  a  a>'-plane  in  which  the  point 
w'  =  oo  corresponds  to  this  point 
of  intersection,  a  triangle  bounded 
by  straight  lines  in  the  o/'-plane 
will  then  correspond  to  the  given 
triangle  of  the  w-plane ;  but  this 
is  simply  the  previous  case  already 
FIG.  48  discussed. 

II.    If  i/t+i/m  +  i/n>i,  we 

transfer  the  triangle  to  the  sphere  by  stereographic  projection  ; 
it  can  then  be  shown  geometrically  that  the  planes  of  the  three 
bounding  circles  intersect  in  a  point  inside  of  the  sphere.  We 
can  now  find  infinitely  many  collineations  of  space  of  the  kind 
spoken  of  in  §  16.  determined  by  linear  transformations  of  the 
w-variables,  which  transfer  the  above  point  of  intersection  to  the 
center  of  the  sphere  ;  if  we  assume  any  one  of  these,  the  triangle 
under  consideration  is  transformed  into  a  "  spherical  triangle  " 
(in  the  ordinary  sense  of  that  word)  which  is  bounded  by  arcs 
of  three  great  circles  of  the  sphere,  and  the  reflections  on  the 
sides  of  the  triangle  defined  in  the  previous  paragraph  are 
thus  converted  into  reflections  with  reference  to  the  planes 
of  these  sides  (cf.  XI,  §  13)  in  the  usual,  optical  sense  of  the 
word  reflection.  Two  successive  reflections  of  this  kind  are 
together  equivalent  to  a  rotation  of  the  sphere  about  the  line 
of  intersection  of  the  two  planes  through  twice  the  angle 
which  these  planes  make  with  each  other.  Therefore  the  figure 
formed  from  the  alternately  symmetric  and  congruent  repetitions 
of  the  original  triangle  must  have  the  property  that  it  is  trans.- 


§  74-   MAP  OF  SPHERICAL  TRIANGLE  ON  HALF-PLANE      41! 

formed  into  itself  by  a  definite  rotation  of  the  sphere  about  its 
center. 

The  inequality  (II)  is  satisfied  by  integral  values  of  /,  ;;/,  n  in 
only  the  following  ways  : 

1.  /=/;/  =  2,  «  arbitrary, 

2.  /=  2,  ;;/  =  3,  #  =  3,  4,  or  5  ; 
the  case                         /—  2,  ;;/  =  3,  ;/  =  3 

will  be  discussed  somewhat  in  detail. 

The  spherical  excess  of  a  triangle  having  the  angles,  7r/2,7r/3, 

7T/3    iS 

(l)  7T/2  +  7T/3  +  7T/3  -  7T  =   7T/6  J 

its  area  is  accordingly  equal  to  one  twenty-fourth  of  the  total  sur 
face  of  the  sphere.  When  it  is  therefore  possible  to  cover  the 
whole  surface  of  the  sphere  once  without  gaps  by  alternately 
symmetric  and  congruent  repetitions  of  the  given  triangle,  we 
shall  need  exactly  twenty-four  such  triangles  for  this  purpose. 
In  fact  the  sphere  is  so  covered  by  dividing  each  face  of  a  regu 
lar  tetrahedron  into  six  triangles  by  drawing  the  medians  in  each 
face,  and  then  projecting  the  triangles  so  obtained  from  the  center 
of  the  tetrahedron  upon  the  surface  of  the  circumscribing  sphere. 
When  such  a  triangle  is  mapped  upon  a  half-plane  in  such  a 
way  that  its  vertices  correspond  to  the  points  z  =  o,  i,  oo  respec 
tively,  the  function  z  of  w  by  which  the  mapping  is  accomplished 
must  have  the  following  properties  (its  existence  always  pre 
supposed)  : 

1.  At  all  points  w  which  are  not  vertices  of  the  triangle,  the 
function  must  be  regular  and  have  a  derivative  different  from 
zero. 

2.  w  —  WQ  must  be  a  regular  function  of  ~\/z  at  the  vertices 
of  the  triangle  WQ  which  correspond  to  the  point  2  =  0,  since  an 
angle  7r/2  on  the  ^-sphere  here  corresponds  to  an  angle  ?r  of 


412       VI.  GENERAL  THEORY  OF  FUNCTIONS 

the  2-sphere  ;  z  is  therefore  a  regular  function  of  w  which  has  a 
zero  of  order  two  at  WQ. 

3.  w  —  wl  must  be  a  regular  function  of  ~\Jz—  i  at  the  ver 
tices  of  the  triangle  w^  which  correspond  to  the  point  z=i,  and 
therefore  z—  i  is  a  regular  function  of  w  which  has  a  zero  of 
order  three  at  wv 

4.  w—w^  must  be  a  regular  function  of  z~^  at  the  vertices  of 
the  triangle  w^  which  correspond  to  the  point  z  =oc,  and  there 
fore  z  is  a  function  of  w  which  has  a  pole  of  order  three  at  w^. 

Accordingly,  z  is  a  function  of  w  which  is  regular  over  the 
whole  w-sphere  with  the  exception  of  particular  poles,  that  is, 
according  to  VI,  §  44  it  is  a  rational  function  of  w.  As  such  it  is 
already  determined  by  the  properties  i,  3,  4,  except  as  to  a  con 
stant  factor ;  the  problem  is  then  solved  when  we  have  so  deter 
mined  this  constant  factor  that  the  property  2  is  also  satisfied. 

The  middle  points  of  the  edges  of  a  tetrahedron  are  the  ver 
tices  of  a  regular  octahedron  ;  we  can  then  think  of  them  as  so 
arranged  that  the  points  w§  corresponding  to  them  on  the  sphere 

fall  at  .    . 

o,     oo,      -f-  i,      +  i,      —  i,      —  /, 

and  are  therefore  (excepting  w  —  oo)  the  roots  of  the  equation  : 
(2)  f,(w)  =w(w*-  i)=o. 

The  vertices  and  the  middle  points  of  the  sides  of  the  tetrahedron 
give  then  points  on  the  sphere  all  three  of  whose  space  coordi 
nates  £,  ry,  £  —  4  (cf.  §  13)  have  the  absolute  value  -  — ;  we  may 

2V3    _ 

suppose  that  the  former  have  an  even  number  of  negative  coor 
dinates  and  that  the  latter  have  an  uneven  number  of  such  coor 
dinates.  Then  the  arguments  W<L  of  the  first  [(6),  §  13]  become : 

i  +  /  i  +/  i  —  /  i  —  / 

v^-i'       V3-i'    Va  +  i'       yi-M 


§  74-    MAP  OF  SPHERICAL  TRIANGLE  ON  HALF-PLANE      413 

that  is,  the  roots  of  the  equation  : 

(3)  /s(a')  =  o>4  —  2  /  V3  w1  +  i  =  o  ; 

the  arguments  wx  of  the  last  become  : 

i  —  /  i  —  /  i  +  i  i  -f*' 


that  is,  the  roots  of  the  equation  : 

(4)  /3  (w)  =  w*  +  2  /  V  3  a/2  4-  i  =  o. 

A  rational  function  z  —  i  of  w  which  satisfies  the  conditions  i, 
3,  4  is  therefore  : 


w4  +  2  i  V3  w*  +  i 

in  order  to  satisfy  also  the  relation  (2)  we  must  have  two  coeffi 
cients  a,  b  satisfying  the  identity : 

We  find : 
(6)  /33  -/23  =  6  (X  +  i)2  2  /  V3  wz  +2(21  V3  zc/2)3 

=  12  V3  ?w2  [(w4  +  ! )2  —  4  z^4]  =  1 2  V 3  if?. 


Hence,  the  desired  function  b\  which  the  given  triangle  bounded 
by  arcs  of  circles  is  mapped  upon  the  half-plane  is : 

(7)  »=»V3*-?='-^ 

/3  /3 

further,  this  function  has  a  two-fold  zero  at  infinity  also  (which 
was  not  considered).* 

*  Concerning  the  case  II  cf.  F.  KLEIN,  Vorlesungen  iiber  das  Ikosaeder,  Leip 
zig,  1884. 


414  VI.    GENERAL  THEORY   OF   FUNCTIONS 

III.  The  third  case  i //  +  i /m  -f  i /n  <  i  leads  to  transcen 
dental  automorphic  functions  ;  we  cannot  enter  into  a  further 
discussion  of  this  case  since  the  object  of  this  introduction  has 
been  obtained,  coming  as  we  now  have  to  the  threshold  of  that 
province  in  the  theory  of  functions  where  some  of  the  most 
appreciated  present-day  problems  are  to  be  found. 

MISCELLANEOUS   EXAMPLES 

1.  Prove  that  every  function  w  =f(z)  determines  a  transfor 
mation  which  leaves  angles  unchanged  over  any  region  through 
out   which  f(z)    is    regular.     What    peculiarity   occurs   in    the 
neighborhood  of  a  branch-point  of  f(z)  ? 

2.  Construct  the  RIEMANN'S  surface  for  the   inverse  of  the 
function  w  =  z4  +  &  and  find  the  images  of  its  sheets. 

3.  If  f(z)  is  analytic  throughout  a  certain  simply  connected 
region,  prove  that 


= 

Ja 


is  also  analytic  there,  a  being  a  fixed  point  of  the  region. 

4.  Show  that  if  f(z)  is  analytic  in  a  certain  region  S  and  if 
f(z)  vanishes  at  every  point  of  S,  then/(z)  is  a  constant. 

/y*** 

5.  By  taking  the  integral    I  —  dz 

*J    z 

along   a    suitable    contour    and    applying    CAUCHY'S    integral 
theorem,  obtain  the  formula 


sin 

x 


and  hence 


rsin  x 
x 


MISCELLANEOUS    EXAMPLES  415 

6.    By  means  of  the  integral 


taken  along  a  suitable  path,  show  that 


Jo 


and  hence  by  means  of  the  same  integral  taken  along  another 
path,  show  that  the  value  of  these  integrals  is  -y-- 

7.  Prove  that,  if  /(z)  is  analytic  at  the  point  z  =  a  and/'(V) 
does  not  vanish,  then  the  equation 

w=f(z) 

can  be  solved  for  z,  and  establish  the  essential  properties  of  the 
solution. 

8.  If  <£(/)  is  a  function  of  t  defined  along  a  regular  curve  C 
in  the   complex  /-plane,  and    if    <£(/)  is  continuous    along  this 
curve,  discuss  the  function  of  z  defined  by  the  integral: 


JG 


G  t—z 

9.  Deduce  CAUCHY'S  integral  formula.  Name  some  of  the 
most  important  theorems  that  are  proven  by  means  of  this 
formula,  and  also  some  that  follow  indirectly  from  it. 

10.    Obtain   an   expression   for  |  sin  z  |   in  terms  of  x  and  y, 
where  z  =  x  +  iy. 

Hence,  discuss  the  convergence  of  the  series 

sin  z  .  sin  2  z  .  sin  3  z  . 
--  1  --  -  --  1  --  ;  --  h  "•• 

5  52  53 

For  what  values  of  z  does   this  series    represent   an   analytic 
function  ? 


4l6       VI.  GENERAL  THEORY  OF  FUNCTIONS 

11.  Regarding  the  function  f(z)  it  is  known  that  its  pure 
imaginary  part  is  never  negative  when  z  lies  in  the  neighbor 
hood  of  the  point  a  ;  while  the  function  <j>(z)  is  in  absolute  value 
greater  than  1/2  for  such  values  of  z.  Both  functions  are  single- 
valued  and  analytic  near  a  with  the  exception  of  the  point  a 
itself,  at  which  they  are  not  defined.  What  can  you  say  about 
the  character  of  the  function 


in  this  neighborhood? 

12.  If  a  function  is  analytic  in  the  entire  plane  and  becomes 
infinite  at  infinity,  will  it  necessarily  vanish  for  some  value  of  z? 

13.  Discuss  the  linear  transformation  of  the  ARGAND  plane 
into  itself  when  the  fixed  points  are  distinct  and  finite. 

14.  Show  that  to  every  rotation  of  a  sphere  about  a  diameter, 
corresponds  a  linear  transformation  of  the  plane  of  stereographic 
projection. 

15.  What    singularities    may    an    algebraic    function    have? 
Prove  your  answer  to  be  correct. 

.16.    State    carefully  a  sufficient   condition    that    an    analytic 
function  be  algebraic. 

17.    Discuss  the  function  defined  by  the  integral 

dz 


w 


-f; 


On  what  region  of  the  w-plane  does  this  function  map  the  upper 
half  of  the  2-plane  ? 

18.  How  would  you  prove  that  every  algebraic  equation  has 
a  root? 

19.  Give  two  definitions  of  the  function  e*  for  complex  values 
of  the  exponent. 


MISCELLANEOUS   EXAMPLES  417 

Restricting  yourself  to  one  of  these  definitions,  show  that  the 
function  is  analytic  and  satisfies  the  functional  relation  : 


20.  State  and  prove  the  theorem  about  the   inverse  of   an 
analytic  function  being  an  analytic  function. 

21.  Prove  that  a  function  f(z)  which  is  analytic  in  all  finite 
points  of  the  z-plane,  and  which  remains  finite  for  the  whole 
plane  is  a  constant. 

22.  The  functions/!  (2),  f»(z)  are  both  analytic  throughout  a 
region  Z"  having  an  isolated  boundary  point  z  =  a,  and  they  have 
poles  at  the  point  a.     What  can  you  say  concerning  the  order  of 
the  poles  of  the  function 

^«  =/.«+/*« 

at  the  point  z  =  a? 

23.  Show  that,  if  a  function  f(z)  is  analytic  throughout  a 
region   T,  one  of  whose  boundary  points  is  the  isolated  point 
z  =  a,  and  if  f(z)  remains  finite  in  the  neighborhood  of  a,  then 

f(z)  approaches  a  limit  when  z  approaches  a. 

24.  A  regular  hexagon  is  reflected  on  its  sides  ;   show  that 
a  +  bv  +  rf  represents  the  vertices  of  the  resulting  configuration  ; 
\ia  +  b  +  c=±  i   (a,  b,  c  integers),  what  particular  hexagon  is 
it?     Discuss  the  case  for  a  +  b  +  c—  ±  a.     Is  the  plane  covered 
simply  or  multiply? 


25.    What  does 


i      i 
a     b 


=  o  mean  if  a,  b,  c ;  x,  y,  z  are  sets 


x    y 

of  points  in  the  plane  ? 

26.  Prove  that  if  a  function  is  analytic  throughout  a  region, 
and  its  vanishing  points  there  are  not  isolated  from  one  another, 
the  function  must  vanish  at  every  point  of  the  region. 


41  8  VI.    GENERAL  THEORY  OF   FUNCTIONS 

What  is  the  importance  of  this  theorem  when  we  come  to  the 
question  of  extending  the  definition  of  functions  from  real  to 
complex  values  of  the  argument?  Illustrate  by  means  of  the 
functions  e*,  sin  z. 

27.  State  accurately  the  three  definitions  of  analytic  functions 
which   depend   respectively  on   the    process   of  differentiation, 
integration,  and  development  in  series. 

Adopting  whichever  of  these  definitions  you  choose,  state  ac 
curately  and  prove  the  theorem  which  says  that  a  uniformly  con 
vergent  series  represents  an  analytic  function. 

28.  If  f(z)  has  at  each  point  of  a  simply  connected  region  n 
distinct  values,  each  of  which  varies  continuously  with  z,  prove 
that  if  the  point  z  describes  any  closed  contour  in  this  region, 
none  of  the  values  oif(z)  will  be  interchanged. 

What  information  does  this  theorem  give  us  concerning  the 
RIEMANN'S  surface  of  the  two-valued  functions 


29.  Construct  the  RIEMANN'S  surface  for  the  inverse  of  the 

function 

z  =  3  w*  -f  4  w3. 

30.  Let  a  be  a  point  within  a  certain  two-dimensional  region 
B,  and  let  /(z)  be  a  function  single-valued  and  continuous  at 
every  point  of  B,  which  vanishes  at  a.     Prove   that  if  /(z)  is 
known  to  be  analytic  at  every  point  of  B  except  a,  it  must  also 
be  analytic  at  a. 

HINT.  —  Use  that  definition  of  an  analytic  function  which  depends  upon  the 
process  of  integration. 

If  <f>(z)  is  analytic  at  every  point  of  B  except  a,  at  which 
point  it  is  not  denned,  and  if  <j>  (z)  does  not  become  infinite  as 


MISCELLANEOUS    EXAMPLES  419 

we  approach  a,  prove  that  it  is  possible  to  define  <f>  at  the  point 
a  in  such  a  way  that  <j>(z)  is  analytic  at  a. 

HINT.  —  Consider  the  function  (z  —  a)0(z). 

31.  Is  z  =  o  a  branch-point  for  the  function  w  of  z  defined  by 
the  equation  ^  =  ^  _  ^  ? 

Is  z  =&  a.  branch  point ?  Construct  the  RIEMANN'S  surface  for 
the  function. 

32.  The  function  f(z)  has  a   branch-point  of   the  (g  —  i)st 
order  in  z  =  a.     When  is  f(z)  said  to  have  a  pole  in  a  ?     Define 
the  order  of  the  pole. 

Will  the  integral  C*  f,  \, 

J  f(z)"2 

necessarily  have  a  pole  in  a  ?     State  precisely  the  condition. 

33.  If   the  analytic  function  w  =f(z)  has  a  branch-point  of 
finite  order  in  z  —  a,  what  is  the  condition  that  the  neighborhood 
of  a  be  mapped  on  a  single-leaved  neighborhood  of  the  point 
w  =f(a)  ?     Discuss  both  the  case  that/(#)  is  finite  and  the  case 

/(«)=». 

34.  Prove  that  if  z  =  x  +  /)',  the  function 

/(z)  =  e1  (cos  y  +  i  sin  y) 

is  an  analytic  function  of  z.  Has  this  function  any  singular 
points?  If  so,  what  are  they?  Are  there  any  points  of 
the  s-plane  where,  in  the  transformation  to  the  o/-plane,  w  =/(z) 
fails  to  be  conformal  ?  If  so,  what  are  they  ? 

What  are  the  images  in  the  ay-plane 

(a)  of  the  lines  parallel  to  the  axis  of  reals  in  the  z-plane ; 

(fr)  of  the  lines  parallel  to  the  axis  of  imaginaries  in  the 
2-plane  ? 


42O       VI.  GENERAL  THEORY  OF  FUNCTIONS 

What  happens  to  the  image  of  a  strip  in  the  s-plane,  bounded 
by  two  lines  parallel  to  the  axis  of  reals,  as  the  breadth  of  this 
strip  increases  indefinitely? 

What  can  be  inferred  from  the  result  you  have  just  found  con 
cerning  the  inverse  of  the  f  unction /(z)  ? 

The  following  four  are  simple  examples  of  conformal  representa 
tion  due  to  SCHWARZ,  Werke,  Vol.  II,  p.  148. 

35.  A  region  bounded  by  two  arcs  of  circles  through  the 
points  z1}  s2  in  the  s-plane  is  mapped  on  half  of  the  w-plane  by 

the  function  /   _  v  X1/A 

w  •=• 


where  A.TT  is  the  angle  at  which  the  arcs  intersect. 

36.  A  region  bounded  by  three  arcs  of  circles  which  intersect 
at  angles  w/2,  77/2,  XTT  is  transformed  by 

a/ =[(*-*!)/(* -**)]1'* 

into  a  semi-circle,  where  z1}  z2  are  the  points  of  intersection  of 
those  two  arcs  which  include  the  angle  X?r(A^o). 

A  special  case  of  the  above  region  is  the  sector  of  a  circle. 
For  this  22==°°?  and  the  transformation  is  replaced  by 

w  =  (z-  *!)1/A. 

37.  If  in  the  preceding  example  X  vanishes,  the  transforma 
tions  which  convert  the  triangle  bounded  by  arcs  of  circles  into 
a  sector  are  of  a  different  character.     Let  Zi  be  the  point  at 
which  the  two  arcs  touch ;  the  remaining  arc  produced  will  pass 
through  o.     If  ATT  be  the  angle  which  the  real  axis  makes  with 
the  tangent  at  zl5  the  transformation 

w  =  ex^l(z  —  zj 

is  equivalent  to  a  turn  of  the  tangent  through  an  angle  A.TT  (thus 
becoming  parallel  to  the  axis)  followed  by  a  quasi-inversion  with 


MISCELLANEOUS   EXAMPLES 


regard  to  zl  (that  is,  a  combination  of  reflection  and  inversion  — 
a  term  due  to  CAYLEY).  The  resulting  region  is  bounded  by  two 
lines  parallel  to  the  real  axis  in  the  ay-plane,  and  by  part 
of  a  line  parallel  to  the  axis  of  imaginaries.  The  further 
transformation 


changes  the  two  parallel  straight  lines  into  two  straight  lines 
through  a  point,  and  the  remaining  straight  line  into  an  arc  of 
a  circle  with  this  point  as  center.  The  resulting  region  is  a  cir 
cular  sector. 

38.  The  transformation  w  =  (z-i)/(z  +  i)  determines  a  con- 
formal  representation  of  the  positive  half  of  the  s-plane  upon  a 
circle  in  the  a/-plane  whose  center  is  at  the  origin  and  whose 
radius  is  unity. 


INDEX 


(The  numbers  refer  to  pages.) 


A2,  principal  value  of,  309. 

ABEL'S  theorem  on  multiplication  of 
series,  212. 

Abscissa,  12. 

Absolute  convergence  of  a  series,  170; 
of  a  power  series,  202. 

Absolutely  convergent  series,  sum  of  an, 
170. 

Absolute  value,  definition  of,  14;  of  a 
sum,  17. 

Absolute  value  and  amplitude  of  a 
complex  number,  218;  of  an  integral, 
155  ;  of  a  product,  19 ;  of  a  quotient,  21. 

Abteilungsu'cisc  monoton,  140. 

Accumulation  point,  128. 

Addition,  geometrical  representation  of, 
16. 

Addition  and  subtraction  of  number- 
pairs,  3. 

Addition  theorem  for  exponential  func 
tions,  218;  for  trigonometric  functions, 
218. 

Aggregate  of  points,  127. 

Algebra,  an,  of  number-pairs,  3  ;  double, 
3 ;  geometrical  representation  of 
double  algebra,  3. 

Algebraic  addition  theorem,  275. 

Amplitude,  definition  of,  15;  change  in 
value  of,  284 ;  and  absolute  value  of  a 
complex  number,  218;  many-valued- 
ness  of  the,  21 ;  of  a  product,  19 ;  of  a 
quotient,  21;  of  the  double  ratio,  70; 
principal  value  of  the,  284 ;  the,  a 
continuous  and  a  single-valued  func 
tion  of  position,  286,  290,  291,  293 ; 
of  the  logarithm,  296. 

Analysis  situs,  328. 

"Analytic  about,"  "regular  at,"  282. 

Analytic  continuation,  382,  384;  across 
a  line,  402  ;  of  an  integral,  385. 


Analytic  function,  an,  definition  of,  384, 
386. 

Analytic  functions  of  analytic  func 
tions,  397 ;  single-valued,  167. 

Angles,  preserved,  41,  43;  in  stereo- 
graphic  projection,  48 ;  opposite  in 
sense,  42,  43. 

Approximation,  degree  of,  153,  169. 

ARGAND,  12. 

Arithmetic,  general,  of  real  numbers,  i,  2. 

Assemblage  of  points,  127. 

Associative  law,  the,  5,  10,  16. 

Automorphic  function,  definition  of,  84; 
symmetric,  92;  example  of,  106,  113. 

Axis,  of  real  numbers,  13 ;  of  pure 
imaginary  numbers,  13 ;  of  reals, 
reflection  on,  39. 

Behavior  of  a  function  at  infinity,  229. 

Bilinear  transformation,  substitution,  54. 

Binomial  theorem,  the,  221. 

BOCHER,  232,  282. 

BOREL,  235. 

BOREL,  theorem,  the  HEINE-,  162. 

Boundary,  natural,  and  singular  points, 
389,  393  ;  a  natural,  a  line  of  singulari 
ties  is,  391 ;  natural,  not  a  cut,  391 ; 
point  of  a  set  of  points,  137. 

Bounds,  upper,  lower,  127,  128;  least 
upper,  128. 

Branch-cut,  cut,  bridge,  321. 

Branch-point,  definition  of,  292,  321,  395  ; 
examples  of,  324-326;  of  many- 
valued  functions,  393 ;  when  a  pole, 
zero,  essential  singularity,  335 ;  when 
a  singular  point,  395. 

Bridge,  branch-cut,  cut,  321. 


CANTOR,  131 ;   DEDEKIND  axiom,  3. 
Cartographic  modulus,  183. 


423 


424 


INDEX 


Cartography,  47. 

Cassinian  oval,  373. 

CAUCHY,  168,  178,  179,  199,  203,  213, 
267,  271,  401. 

CAUCHY-RIEMAXN  differential  equations, 
176,  179,  181. 

CAUCHY'S  theorems  on  integration,  190, 
195,  199,  247;  on  residues,  236; 
applied  to  Riemann's  surfaces,  333. 

CAYLEY,  43,  421. 

Change  of  amplitude,  284. 

CHAPMAN,  23. 

Circle  of  convergence,  199,  201 ;  of  a 
power  series,  202. 

Circle  transformation,  54,  56;  unit 
circle,  15. 

Circuits  about  the  origin,  288. 

Circular  points  at  infinity,  70. 

Class  of  points,  127. 

Classification  of  integrals  by  disconti 
nuities,  352. 

Closed  segment,  138;  set  of  points,  138. 

Cluster  point,  128. 

Coefficients,  of  a  power  series,  206-210; 
method  of  undetermined,  207. 

Coincidence  of  two  power  series,  207 ; 
of  two  functions,  208,  384. 

Collection  of  points,  127. 

Collinear  points,  three,  26. 

Collineations  of  space,  77,  79,  80,  410. 

Commutative  law,  5,  7,  16. 

"Complete"  equation,  definition  of  a, 
299. 

Complex  function,  170;  continuity  of  a, 
171. 

Complex  numbers,  conjugate,  15,  19; 
geometrical  representation  of,  i,  12; 
multiplication  of,  10;  on  the  helicoid, 
291;  on  the  sphere,  51;  real,  pure 
imaginary  parts  of,  10;  units  of,  8. 

Complex  quantity,  9. 

Condensation  point,  128. 

Confocal  parabolas,  85. 

Conformal  representation,  28,  41,  43, 
182;  breaks  down,  184;  by  singly 
periodic  functions,  224;  by  the  loga 
rithm,  304;  by  tan"1  z,  314;  of  a  tri 
angle  on  the  half -plane,  403,  409;  of 
the  half-plane  on  a  circle,  421 ;  where 
dw/dz  =  o,  395. 


Congruent  figures,  30,  31. 

Conjugate  complex  numbers,  15,  19. 

Connected,  domain,  simply,  142 ;  seg 
ment,  138;  set  of  points,  138. 

Connectivity  of  the  surface  for  ^~z,  328. 

Constant,  definition  of  a,  28;  when  a 
function  is,  230. 

Construction  of  the  RIEMANN'S  surface, 
general,  385. 

Continuation,  analytic,  382,  384. 

Continuity,  127;  at  infinity,  173;  at  a 
point,  131;  at  a  pole,  172;  in  a  do 
main,  145;  of  complex  functions,  171 ; 
of  functions  of  two  variables,  143,  144 ; 
of  a  rational  function,  133,  170,  172; 
on  an  interval,  131 ;  uniform,  132. 

Continuous  function,  a  =  a  power  series, 
203;  continuous  mapping,  148,  151. 

Contour,  rectangular,  158. 

Convergence,  absolute,  of  a  series,  170; 
absolute,  of  a  power  series,  202  ;  circle 
of,  199,  201 ;  of  an  infinite  series,  169, 
1 70 ;  of  a  power  series,  201 ;  of  a 
product  of  two  series,  212;  perma 
nent,  of  a  power  series,  201 ;  uncon 
ditional,  170;  uniform,  155,  156,  200, 
202,  203. 

Convergent  sequence,  limit  of  a,  168. 

Coordinates  of  a  point,  12;   polar,  13. 

Correspondence,  one-to-one,  12,  13,  29, 
127. 

Cosine,    216-221;     a    symmetric    auto- 
morphic  function,  226;   period  of  the, 
222. 
I  Cube  roots  of  unity,  25,  72. 

Curve,  136;  =  a  set  of  points,  136,  138, 
139;  general  definition  of  a,  139,  140; 
parametric  representation  of  a,  139; 
simple,  140. 

Curvilinear  integral,  156,  158,  189. 

Cut,  321;  Dedekind,  partition,  128; 
not  a  natural  boundary,  39 1- 

DARBOUX,  152,  402. 

Decomposition  into  partial  fractions,  268. 
DEDEKIND-CANTOR  axiom,  3. 
DEDEKIND  cut,  partition,  128. 
Definitely  infinite,  255,  256. 
Deformation  of  the  surface  for  -^/~z,  329. 
Degenerate  transformation,  55. 


INDEX 


425 


Degree  of  approximation,  153,  169;  of  a 
function,  99,  102 ;  of  a  rational  func 
tion,  102. 

DE  MOIVRE'S  theorem,  generalization  of, 

309- 

Dense  in  itself,  a  set  of  points,  138. 

Derivatives,  definition,  148;  higher,  suc 
cessive,  179 ;  independent  of  approach, 

175,  176,  178;    logarithmic,  240,  374; 
of  a  determinant,   166;    of  a  rational 
function,  174;   of  an  integral,  206;   of 
a  power  series,  149,  203;  partial,  150; 
progressive,      regressive,      163,      165; 
successive,    of    a    power    series,    204; 
successive,  of  a  regular  function,  204; 
term  by  term,  149,  156. 

Derived,  function,  149;    sets  of  points, 

131- 

DESCARTES,  12. 
Determinant,  derivative  of  a,  166;   of  a 

transformation,  55,  65. 
Development,   in   a   circular   ring,    249, 

253  ;  near  a  pole,  229 ;   in  a  strip,  257  ; 

in  a  power  series,  199,  206  ;  in  a  power 

series,  unique,  207. 

Diametral,  two  complex  numbers,  53,  So. 
Differential  equations,  CAUCHY-RIEMAXX, 

176,  179,  181 ;  partial,  280. 
Differentiation,   the  inverse  of  integra 
tion,  155,  192. 

Direct  similarity  transformation,  35 ; 
direct  circle  transformation,  56. 

Discontinuity,  removable,  103,  228,  252. 

Discontinuous,  finite  group,  113. 

Displacement  (a  translation),  30,  33. 

Distributive  law,  7. 

Division  by  zero,  45 ;  of  complex  num 
bers,  20,  21 ;  of  a  line  segment,  har 
monic,  6 1 ;  of  number- pairs,  6. 

Domain,  140;  improper,  141;  mapped 
continuously,  148,  151;  simply  con 
nected,  142. 

Double  algebra  and  its  geometrical  repre 
sentation,  3. 

Double  ratio,  amplitude  of,  70 ;  invariant, 
65,  72  ;  of  four  points,  69,  70. 

e2,  216 ;  is  a  symmetric  automorphic  func 
tion,  225;  fundamental  region  of,  225. 
Electrical  potential,  187. 


"Element"  of  the  analytic  function,  384. 
Elliptic  linear  substitution,  94,  95. 
Equation,  a  complete,  299 ;  of  nth  degree 

has  n  roots,  242 ;  has  no  more  than  n 

roots,  99. 

Equator  on  sphere,  51. 
Equatorial  plane,  reflection  on,  51. 
Equianharmonic  points,  72. 
Equipotential,  lines  of,  187. 
Essential  singularity,   definition  of,  389, 

300,  392 ;   at  a  branch-point,  335. 
EULER,  Si,  217. 
EULER'S  relations,  217,  316. 
Even,  odd  functions,  272. 
Expansion  (cf.  also  development) ;    in  a 

circular  ring,   249,    253 ;    in  a  power 

series,  109;  in  a  strip,  257. 
Exponential     function,     216-221,     279; 

addition  theorem  for,  218;   periodicity 

of,  222. 

Finite  discontinuous  group,  113. 

Fixed  (invariant)  points  of  a  transforma 
tion,  33  ;  of  the  linear  fractional  trans 
formation,  57,  64. 

Fldfhfnstu-ckc  (region),  136. 

Flow,  lines  of,  187  ;   of  a  fluid,  185. 

FORSYTE,  142,  261. 

ForRiER's  series,  257. 

Fractional  numbers,  2. 

FRESNEL  integrals,  215. 
i  Function,  automorphic,  84 ;   example  of 

automorphic,  106,  113. 
I  Function,  behavior  at  infinity,  104,  229. 

Function,  complex,  170. 
;  Function,  constant,  when,  230. 

Function,  continuity  of  a  rational,  133, 
170,  172. 

Function,  continuous  where  finite,  255. 

Function,  continuous  =  a  power  series, 
203- 

Function,  definition  and  history  of,  28, 
29,  178. 

Function,  definition  of  an  analytic,  384, 
386. 

Function,  definition  of  a  rational,  28,  29, 
167,  231,  392. 

Function,  definition  of  a  regular,  178. 

Function,  degree  of  a  rational,  103. 

Function,  derivative  of  a  rational,   174. 


426 


INDEX 


Function,  "element"  of  the,  384. 
Function,  even,  odd,  272. 
Function,  exponential,  216-221,  279. 
Function,  generalization  of  the  transcen 
dental,  216. 

Function,  holomorphic,  252. 
Function,  infinite  value  of  a,  173. 
Function,  linear  fractional,  54. 
Function,  linear  integral,  32-38. 
Function,  map  of  a  rational,  173. 
Function,  mapping  with  a  regular,  182. 
Function,  monogenic,  178. 
Function,  w-valued  at  a  point,  242. 
Function,  order  of  the  rational,  102,  103. 
Function,  periodic,  84,  222,  284. 
Function,    rational,    the,    indeterminate, 

45,  103,  174,  228. 
Function,  rational  integral,  29,  98,  231, 

233,  392. 

Function,  rational,  of  z  and  s  =  Vz,  331. 
Function,  reciprocal  of  a  rational,   172, 

173. 

Function,  regular  =  a  power  series,  203. 
Function,  regular,  expanded  in  a  power 

series,  199. 
Function,  regular,  near  a  singular  point, 

227,  247,  254. 
Function,    regular,    on    the    RIEMANN'S 

surface,  335. 

Function,  single- valued  analytic,  167. 
Function,    single-valued    on    the    RIE 
MANN'S  surface,  333. 
Function,   single-valued  on  the   sphere, 

389- 

Function,  singly  periodic,  224,  268,  273. 
Function,    successive    derivatives    of    a 

regular,  204. 
Function,  sum  of  all  orders  of  a  rational, 

106,  242. 

Function,  symmetric  automorphic,  92. 
Function,  transcendental,  124,  167,  282, 

392. 
Function,   transcendental   integral,    201, 

392. 
Function,    transcendental     integral,     at 

infinity,  233. 

Function,  trigonometric,  216-221. 
Function,  value  of  a  periodic,  233. 
Function,  value  of  a  rational  at  infinity, 

105.  173- 


Function,  value  of  a  regular  in  a  domain, 
196. 

Function,  when  co  is  a  pole  of  a  rational, 
105. 

Function,  when  oo  is  a  zero  of  a  rational, 
106. 

Function,  with  removable  discontinuity, 
228. 

Functions,  zeros  and  poles  of  a  rational, 
242. 

Function,  \/z,  the,  365  ;  s2  =  i  —  z3,  the, 
368. 

Function,  w  =  z2,  the,  82. 

Function,  w  =  zn,  the,  87 ;  is  automor 
phic,  88;  determines  a  cyclic  group, 
90;  fundamental  region  for,  92. 

Function,  z  =  w  +  i  Vi  —  w2,  the,  355. 

Functions,  analytic,  of  analytic  func 
tions,  397. 

Functions,  coincidence  of,  208,  384. 

Functions,  hyperbolic,  217,  219-221. 

Functions,  many-valued,  284. 

Functions,  of  a  real  variable,  127. 

Functions,  of  ~v  (z  —  a)/ (z  —  b)  and 
V(z  —  a)  (z  -  b),  342,  368. 

Functions,  rational,  of  z  and  <r  = 
V(z  —  a)  (z  —  b),  344. 

Fundamental  laws,  the,  i,  5,  7,  10. 

Fundamental  operations,  22,  23. 

Fundamental  region,  definition  of,  84. 

Fundamental  region,  for  cz,  225. 

Fundamental  region  for  w  =  zn,  91. 

Fundamental  theorem  of  algebra,  229, 
232,  240. 

GAUSS,  9,  12,  43,  308. 

GAUSS'S  series,  214. 

Generalization  of  transcendental  func 
tions,  216. 

Generalization  of  DE  MOIVRE'S  theorem, 
309- 

Geometrical  representation  of  addition, 
1 6 ;  of  complex  numbers  on  the  sphere, 
51;  of  subtraction,  18;  of  multiplica 
tion,  18-20. 

GMEINER,  A.,  and  O.  STOLZ,  22. 

GORDAN,  232. 

GOURSAT,   190. 

GOURSAT-HEDRICK,  182,  232. 


INDEX 


427 


GREEN'S  theorem,  280. 

Group,    cyclic,    go;     definition    of,    57; 

finite  discontinuous,  93,  113;  invariant 

of  a,  72. 

HANKEL,  H.,  23. 

HARKNESS  and  MORLEY,  186,  188,  280, 

384- 

Harmonic,  division  of  a  line  segment,  61 ; 
points,  72. 

HEINE,  132. 

HEINE-BOREL  theorem,  162. 

Helicoidal  surface,  200;  complex  num 
bers  on,  a,  291 ;  pitch  of  the,  292. 

HERZ,  308. 

Higher  derivatives  (cf.  also  successive 
derivatives),  179. 

HOFFMAN,  FR.,  329. 

Holomorphic  function,  252. 

Hyperbolic  functions,  217,  219-221. 

Hyperbolic  linear  substitution,  94-98. 

Hypergeometric  series,  214. 

Images,  reflected,  408. 

Imaginary  axis,  13  ;   imaginary  numbers 

as  number-pairs,  3  ;   imaginary  part  of 

complex  numbers,  10. 
Improper  domain,  141;    path,  141. 
Independent  of  the  path,  the  derivative 

is,  175,  176,  178;   the  integral  is,  191. 
Indeterminate  value  of  a  function,  45, 

103,  174,  228. 
Inferior  limit,  130. 
Infinite,  definitely,  255,  256. 
Infinite  series,  convergence  of,  169,  170; 

sum  of,  169 ;   of  partial  fractions,  263. 
Infinite  value,  fundamental  laws  for,  46; 

of  a  complex  variable,  45,  47 ;    of  a 

function,  173  ;   on  the  sphere,  51. 
Infinitesimal,  definition  of,  182  ;  triangle, 

183- 
Infinity,  as  an  inner  point,  238 ;  behavior 

of  a  function  at,   229;    circular  points 

at,  70;    corresponds  to  zero,  45,  104; 

v-fold    of    a    rational    function,    102 ; 

point  at,  45,  48;   regarded  as  i/o,  45; 

two  views  of,  1 73 ;  value  of  a  function 

at,  105,  173- 

Inner  point  of  a  set,  137. 
Integers  of  arithmetic,  i,  2. 


Integral,  absolute  value  of,  155;  along 
a  closed  curve,  100,  195,  196;  analytic 
continuation  of  an,  385 ;  classified  by 
discontinuities,  352 ;  curvilinear,  156, 
158,  189;  definition  of  a  definite,  151 ; 
derivative  of  an,  206;  equal  to  zero, 
190,  191,  193,  195. 

Integral  function,  linear,  32-38. 

Integral  function,  rational,  29,  98,  231, 

233- 

Integral  function,  transcendental,  defini 
tion,  201,  392. 

Integral  function,  transcendental,  as  a 
product,  375. 

Integral  function,  transcendental,  at 
infinity,  233. 

Integral,  independent  of  the  path,  191 ; 
of  a  rational  function,  351 ;  of  a  regu 
lar  function,  188;  transformation, 
linear,  32-38;  upper  limit  of  an,  189; 
upper,  lower,  152 ;  with  logarithmic 
discontinuities,  353. 

Integrals,  FRESNEL,  215. 

Integrand,  154. 

Integration,  arithmetic  definition  of, 
151;  between  complex  limits,  1 88 ; 
CAUCHY'S  theorem  on,  100,  199;  in 
verse  of  differentiation,  155,  192 ;  of 
a  series  term  by  term,  155,  200;  path 
of,  189. 

Interval,  partition  of  an,  151;  segment, 
127. 

Invariant,  double  ratio  is,  65  ;  of  a  group, 
72 ;  (fixed)  points,  33,  57,  64. 

Inverse  circular  and  the  logarithmic  func 
tions,  365. 

Inverse  of  the  logarithm,  305,  306. 

Inverse  sine,  defn.,  360. 

Inverse  tangent,  defn.,  313  ;  poles  of  the, 
315- 

Inverse  transformation,  the,  57,  114. 

Involutoric  transformation,  38,  40,  75. 

Irrational  numbers,  2,  127,  128. 

Isogonal  representation,  43. 

Isolated  point  of  a  set,  137 ;  isolated 
singular  point,  227,  247,  254,  389. 

Isometric,  isothermal  system  of  curves, 
185. 

JORDAN,  152. 


428 


INDEX 


KlRCHHOFF,  96. 

KLEIN,  94,  119,  413. 

LAGRANGE,  308. 

LAPLACE,  180. 

LAURENT'S  series,  247-251,  254. 

LAURENT'S  theorem  applied  to  the  sur 
face  for  Vz,  339. 

Law,  the  associative,  5,  10;  the  commu 
tative,  5,  7  ;  the  distributive,  7. 

Laws,  the  fundamental,  i,  5,  7,  10. 

Least  upper  bound,  128. 

LENNES,  VEBLEN  and,  28,  131,  140,  162. 

Level,  lines  of,  187. 

LIE,  S.,  23. 

Limit  of  a  convergent  sequence,  168;  of 
a  complex  function,  167,  171;  of  a 
function  of  two  real  variables,  144; 
superior,  inferior,  130;  upper,  of  an 
integral,  189;  uniform  approach  to, 
132,  141,  155. 

Limit  points,  127,  128;  in  the  plane,  137. 

Line  of  singularities,  391 ;  is  a  natural 
boundary,  391. 

Linear  fractional  transformation,  54. 

Linear  integral  function,  transformation, 
32-38. 

Linear  substitution,  transformation,  54. 

Linear  transformation  into  itself,  a,  84; 
on  the  sphere,  77. 

Lines,  as  sets  of  points,  138 ;  niveau,  187  ; 
of  equipotential,  187;  of  flow,  187;  of 
level,  187. 

Logarithm,  amplitude  of  the,  296;  a 
many-valued  function,  297  ;  a  regular 
function,  297 ;  conformal  representa 
tion  by  the,  304;  definition  of  the, 
294 ;  expansion  of  the,  in  a  TAYLOR'S 
series,  297,  298;  inverse  of  the,  305, 
306;  natural,  294;  of  a  real  positive, 
negative  number,  297  ;  principal  value 
of  the,  295,  297,  304;  real,  296,  304; 
RIEMANN'S  surface  of  the,  297. 

Logarithmic,  derivative,  240,  374;  dis 
continuities  of  integrals,  353  ;  and  in 
verse  circular  functions,  365  ;  residue, 

354- 

Lower  bound  =  upper  integral,  152; 
lower  integral,  152;  lower  limit,  130; 
lower,  upper  bounds,  127,  128. 


Loxodrome  or  rhumb  line,  307. 
Loxodromic  linear  substitution,  94. 

Maclaurin's    development    in    a    power 

series,  206,  207. 
Many-fold  root,  99. 
Many-valued  functions,  284. 
Many-valuedness  of  the  amplitude,   21. 
Map,  of  a  rational  function,   1 73 ;    of  a 

rectangle  upon  a  circular    ring,    257 ; 

of  surfaces,  construction  of,  reference, 

308 ;    of  the  2-plane,  29 ;    of  a  triangle 

on  the  half -plane,  403,  409. 
Mapping,  general  continuous,   148,   151, 

182  ;    with  preservation  of  angles,  43  ; 

with  a  regular  function,  182 ;    with  w 

=  z2,  83,  85-87 ;  with  w  =  zn,  88. 
MERAY,  Ch.,  168. 
MERCATOR'S  projection,  307. 
Meridians  on  the  sphere,  51. 
Method  of  undetermined  coefficients,  207. 
MITTAG-LEFFLER'S  partial  fractions,  373 ; 

theorem,  262. 
MOBIUS,  54,  72. 
Modulus,  14;  cartographic,  183;  of 

periodicity,  352. 
Monogenic  function,  178. 
Monotonic,  134 ;  partitively,  140. 
MORLEY,  HARKNESS  and,  186,  188,  280, 

384- 

Motions  in  space  as  a  collineation,  80. 

Multiplication,  geometrical  representa 
tion  of,  18-20;  of  complex  numbers, 
10;  of  number-pairs,  6;  of  series,  212. 

Natural  boundaries,  a  line  of  singularities, 
391 ;  and  singular  points,  389,  393. 

Natural  logarithm,  294. 

Negative  numbers,  2. 

Neighborhood,  circular,  rectangular,  137  ; 
of  a  point,  136. 

NEUMANN'S  sphere,  47,  77. 

Niveau  lines,  187. 

Non-collinear,  three  points,  24. 

Non-essential  singular  point  or  pole,  227, 
228,  247,  254,  389,  392. 

Norm,  14,  19. 

nth  root  of  z,  the,  365. 

n-va.lu.ed  function  at  a  point,  242. 

Number  of  zeros  and  poles,  240-242. 


INDEX 


429 


Number-pairs,  addition  and  subtraction 
of,  3-6 ;  as  complex  numbers,  7 ;  as 
"imaginary"  numbers,  3;  division  of, 
6 ;  equality  of,  4 ;  in  the  form  (a  +  bi) 
9;  multiplication  of,  6,  7;  opposite, 
5  ;  units  for,  7. 

Numbers,  a  notation  for  points,  127. 

Numbers,  complex,  geometrical  represen 
tation  of,  i,  12;  multiplication  of,  10; 
units  of,  8. 

Numbers,  "  imaginary,"  as  number-pairs, 

3,  7- 

Numbers,  irrational,  127,  128. 
Numbers,  negative,  fractional,  irrational, 

2. 

Numbers,  opposite  complex,  15. 
Numbers,  real,  general  arithmetic  of,  i, 

2;  combinations  of,  i. 

Odd,  even  function,  272. 

One-to-one  correspondence,    12,    13,   29, 

127. 

Opposite  complex  numbers,  15. 
Order  of  a  rational  function,  102,  103 ; 

sum  of  all,  106,  242. 
Order  at  a  pole,  at  a  zero,  103. 
"Origin"  of  coordinates,  the,  12. 
Orthomorphic  transformation,  43. 
OSGOOD,  25,  142,  186,  232,  384. 

PAINLEVE,  P.,  400. 

Parabolic  linear  substitution,  94-97. 

Parallel  displacement,  30. 

Parallel  translation  in  the  plane,  28,  30. 

Parametric   representation   of    a   curve, 

139- 

Partial  derivative,  150. 
Partial  differential  equations,  280. 
Partial  fractions,  decomposition  into,  268. 
Partial  fractions,  infinite  series  of,  263. 
Partial     fractions,     MITTAG-LEFFLER'S, 

373- 

Partition,  DEDEKIND,  128. 
Partition  of  an  integral,  151. 
Partitively  monotonic,  140. 
"Parts''  of  the  RIEMANN'S  surface,  323. 
Path,  140;  improper,  141. 
Path-curves,  187. 
Path  of  integration,  189. 
Perfect  set  of  points,  138. 


Period  of  trigonometric  and  exponential 

functions,  222,  223. 
Period,  primitive,  222;  strip,  224-226. 
Periodic  function,  84,  222,  282;    singly, 

224,  268,  273;   values  of,  233. 
"Periodicity,  modulus  of,"  352;    of  the 

trig,  and  exp.  functions,  222. 
Permanently    convergent    power    series, 

201. 
PICARD,  235,  267,  278. 

PlERPONT,  3,   128. 

Pitch  of  the  helicoid,  292. 

Plane  RIEMANX'S  surface,  292. 
i  POIXCARE,  98. 

Point,  at  infinity,  45,  48 ;  boundary,  137 ; 
inner,  of  a  set,  137;  isolated,  137; 
limit,  127,  128. 

Points,  aggregate,  assemblage,  class,  col 
lection  of,  127;  diametral,  53,  80; 
equianharmonic,  harmonic,  72  ;  of  the 
plane,  12;  sets  of,  on  a  straight  line, 
127  ;  sets  of,  in  the  plane,  135. 

Polar  coordinates,  13. 

Pole,  at  a  branch-point,  335  ;  at  infinity, 
105;  continuity  at  a,  172  ;  many-fold, 
102 ;  or  nonessential  singular  point, 
227,  247,  254,  389,  392 ;  residue  at  a, 

237- 

Poles  and  zeros,  242,  243 ;  number  of, 
240-242  ;  of  the  inverse  of  tan"1  z,  315. 

Positive  half -plane,  67. 

Potential,  electrical,  187;  velocity,  186, 
187. 

Power  series,  168;  absolute  convergence 
of,  202 ;  a  continuous  function,  203 ; 
a  regular  function,  203  ;  coefficients  of 
a,  206-2 10 ;  coincidence  of,  207  ;  con 
vergence  of,  201 ;  circle  of  convergence 
of,  202  ;  derivative  of,  149,  203  ;  devel 
opment  in,  199,  207 ;  permanently 
convergent,  201 ;  properties  of,  201 ; 
successive  derivatives  of,  204;  TAY 
LOR'S,  MACLAURIN'S,  206;  uniform 
convergence  of,  202,  203. 

Primitive  period,  222. 

Principle  of  reflection,  the,  399. 

Principal  value  of  the  amplitude,  284; 
of  the  logarithm,  295,  297,  304;  of  the 
square  root,  319. 

PRINGSHEIM,  165. 


430 


INDEX 


Product,  absolute  value  of,  ig;  ampli 
tude  of,  19;  form  for  the  sine,  376; 
form  for  a  transcendental  integral 
function,  375 ;  of  two  complex  num 
bers,  unique,  10;  of  two  series,  con 
vergence  of,  212;  WEIERSTRASSIAN, 
373;  when  zero,  n. 

Progressive,  regressive  derivatives,  163, 
165. 

Projection,  MERCATOR'S,  307 ;  stere- 
ographic,  47,  53,  307 ;  angles  pre 
served  in  stereographic,  48. 

Pure  imaginary  numbers,  10. 

Quadruple  numbers,  22. 
Quantity,  complex  (a  complex  number),  g. 
Quotient,  20;   of  two  conjugate  complex 
numbers,  21. 

Radius  vector,  13. 

Rational  function,  behavior  of  a,  at 
infinity,  104;  continuity  of  a,  133,  170, 
172;  definition  of  a,  28,  2g,  167,  231, 
3g2 ;  degree  of,  102  ;  derivative  of  a, 
174;  indeterminate,  45,  103;  many- 
fold  pole  of  a,  102  ;  many-fold  zero  of 
a,  102;  map  of  a,  173;  order  of  a, 
102,  103  ;  pole  of  a,  242  ;  reciprocal  of 
a,  172,  173,  228;  sum  of  all  the  orders 
of  a,  106,  242 ;  value  of  a,  at  infinity, 
105;  when  infinity  is  a  pole,  a  zero 
of  a,  105,  106;  zero  of  a,  242. 

Rational  functions  of  2  and  s  =  \/z, 
331 ;  of  z  and  <r  =  V  (z  —  a)  (z  —  0), 
344- 

Rational  integral  function,  2g,  g8,  231, 
233,  2g2,  3g2 ;  has  w-fold  pole  at  oo, 
242,  3g2  ;  has  n  zeros,  242. 

' '  Rationalizing, "  345. 

Real  axis,  13. 

Real  logarithm,  2g6. 

Real  numbers,  combinations  of,  i ;  gen 
eral  arithmetic  of,  i,  2. 

Real  part  of  complex  numbers,  10. 

Real  variable,  functions  of  a,  127. 

Reciprocal  of  a  complex  number,  21. 

Reciprocal  of  a  rational  function,  172, 
173,  228. 

Reciprocal  radii,  transformation  by,  38, 
41 ;  on  the  sphere,  51. 


Rectangular  contour,  158. 

Reflected  images,  408. 

Reflection  on  axis  of  reals,  3g ;  on  a 
circle,  407 ;  on  equatorial  plane,  52  ; 
on  a  straight  line,  402 ;  on  unit  circle, 
40;  principle  of,  3gg. 

Regressive,  progressive  derivatives,  163, 
165- 

"Regular  at,"  "analytic  about,"  282. 

Regular  function,  definition,  178;  ex 
panded  in  a  power  series,  igg;  map 
ping  with  a,  182  ;  near  a  singular  point, 
227,  247,  254;  on  the  RIEMANN'S  sur 
face,  334;  successive  derivatives  of, 
204;  value  in  a  domain,  ig6. 

Regular  functions,  sums  of,  260. 

Removable  discontinuities,  103,  228, 
252. 

Representation,  conformal,  28,  41,  43, 
182,  184. 

Representation,  geometrical,  of  addition 
and  subtraction,  16;  of  complex  num 
bers,  i,  12;  of  double  algebra,  3;  of 
multiplication,  18-20. 

Representation,  isogonal,  43  ;  of  z-plane, 
2g;  parametric,  i3g;  similar  in  infini 
tesimal  parts,  43. 

Residue  at  a  logarithmic  discontinuity, 
354;  at  a  pole,  237,  241. 

Residues,  237-23g;  CAUCHY'S  theorem 
on,  236. 

Rhumb  line  or  loxodrome,  307. 

RIEMANN,  2g,  47,  168,  178,  i7g,  214. 

RIEMANN'S  surface,  general  construction 
of  the,  385;  of  the  amplitude,  28g;  of 
the  logarithm,  2g7  ;  of  the  square  root, 
3ig;  plane,  2g2,  2g3 ;  "sheets"  of 
the,  2g3. 

Root  of  a  number-pair,  square,  g. 

Roots,  limits  of  the,  100,  101 ;  many- 
fold,  gg;  o$  unity,  25,  72. 

Rotating  and  stretching,  32,  33. 

Rotating  the  plane  into  itself,  31. 

Scale  of  similarity,  183. 
SCHWARZ,  H.  A.,  402,  407,  420. 
Screw  surface,  2go. 

Segment,  127;   closed,  connected,  138. 
Sequence,   convergent,   i2g;    increasing, 
130;   limit  of  convergent,  168. 


INDEX 


431 


Series,  absolute  convergence  of,  170; 
convergence  of,  169,  170,  201 ;  con 
vergence  of  a  product  of  two,  212; 
FOURIER'S,  257  ;  GAUSS'S,  214;  hyper- 
geometric,  214;  of  partial  fractions, 
infinite,  263;  LAURENT'S,  247-251, 
254;  power  (cf.  power  series);  sum 
of  an  infinite,  169. 

Set  of  points,  closed,  connected,  perfect, 
138;  as  lines,  surfaces,  138. 

Sets  of  points,  dense,  138;  derived,  131 ; 
inner  point  of  a,  137 ;  on  a  straight 
line,  127. 

"Sheets"  of  a  RIEMANN'S  surface,  293, 

323- 

Similar  figures  transformed  into  each 
other,  36. 

Similar  triangles,  condition  for,  36. 

Similarity,  scale  of,  183  ;  transformation, 
32,  35- 

Simple  curve,  140. 

Simply  connected  domain,  142. 

Sine,  216-221 ;  is  a  symmetric  automor- 
phic  function,  226;  inverse,  360; 
period  of,  222  ;  product  for  the,  376. 

Single-valued  analytic  functions,  167. 

Single-valued  function  on  the  RIEMANN'S 
surface,  333. 

Singly  periodic  function,  224;  conformal 
representation  by,  224;  general  theo 
rems  on,  2  73 ;  partial  fractions  of, 
268. 

Singularities,  branch-points  are,  when, 
393.  395  J  line  of,  391 ;  line  of,  as  a 
natural  boundary,  391 ;  and  natural 
boundaries,  389,  393  ;  removable,  103, 
228,  252. 

Singular  point,  essential,  389,  390,  392 ; 
isolated,  227,  247,  254,  389;  non- 
essential  or  pole,  227,  228,  389,  392; 
regular  function  near  a,  227,  247,  254. 

Sin"1  w,  360. 

Situs,  analysis,  328. 

Sphere,  47;   NEUMANN'S,  47,  77. 

Square  root,  316-319;  branch-points  of, 
322;  definition,  317;  connectivity  of 
its  surface,  328;  deformation  of  its 
surface,  329;  expansion  in  a  power 
series,  337;  of  a  number-pair,  9; 
principal  value  of,  319;  RIEMANN'S 


surface  of,  319;  a  single- valued  func 
tion  of  position,  322. 

Squares,  indefinitely  small,  185,  187. 

Stereographic  projection,  47,  53,  307. 

Stretching  and  rotating,  32,  33. 

Stretching  the  plane  into  itself,  31. 

STOLZ,  O.,  and  A.  GMEINER,  22. 

Strip,  period,  224-226. 

Subgroup,  57. 

Substitution,  linear  (=  bilinear),  54; 
elliptic,  hyperbolic,  loxodromic,  para 
bolic,  94-98. 

Subtraction,  geometrical  representation 
of,  1 8. 

Successive  derivatives  of  a  power  series, 
of  a  regular  function,  204. 

Sum,  absolute  value  of  a,  17. 

Sum  of  an  absolutely  convergent  series, 
170;  of  an  infinite  series,  169;  of 
regular  functions,  260. 

Superior  limit,  130. 

Surface,  136 ;  as  a  set  of  points,  138 ; 
general  definition  of,  139,  140;  helicoid, 
screw,  200 ;  plane,  RIEMANN'S,  292,  293 ; 
RIEMANN'S  (cf.  RIEMANN'S  surface). 

Tan"1  s,  definition,  313;  mapping  with, 
314;  poles  of  the  inverse  of,  315; 
principal  value  of,  313. 

TAYLOR'S  series,  206,  209,  213,  383;  for 
the  logarithm,  297,  298. 

Term  by  term  differentiation,  149,  156. 

Term  by  term  integration,  155,  200. 

Theorem,  addition,  for  trigonometric  and 
exponential  functions,  218;  binomial, 
221 ;  DE  MOIVRE'S,  a  generalization  of, 
309;  fundamental,  of  algebra,  229,  232, 
240;  GREEN'S,  280;  LAURENT'S,  247, 254; 
LIOUVILLE'S,  230;  MITTAG-LEFFLER'S, 
262 ;  on  integration,  CAUCHY'S,  190, 
195,  199,  247;  on  residues,  CAUCHY'S, 
236. 

Theorems,  general,  on  singly  periodic 
functions,  273. 

THOMAE,  152,  168. 

THOMSON  images,  40. 

Topology,  328. 

Total  variation  of  a  quantity,  175. 

Transcendental  functions,  124,  167,  282, 
392;  generalization  of,  216. 


432 


INDEX 


Transcendental  integral  function,  defini 
tion,  201 ;  as  a  product,  375 ;  at 
infinity,  233,  255. 

Transformation,  bilinear,  54. 

Transformation,  by  rotating  the  sphere, 
52. 

Transformation,  circle,  54,  56. 

Transformation,  degenerate,  55. 

Transformation,  determinant  of  the,  55. 

Transformation,  "direct"  similarity,  35. 

Transformation,  identical,  57. 

Transformation,  into  itself,  linear,  84. 

Transformation,  inverse,  57,  114. 

Transformation,  involutoric,  38,  40,  75. 

Transformation,  linear  (cf.  substitution, 
linear). 

Transformation,  linear  fractional,  54. 

Transformation,  linear  integral,  32-38. 

Transformation  of  the  sphere  into  itself, 

Si- 

Transformation  of  the  plane,  29 ;  by 
reciprocal  radii,  38;  by  rotation,  31; 
by  stretching,  31 ;  by  a  translation,  30. 

Transformation  on  the  sphere,  linear,  77. 

Transformation,  orthomorphic,  43. 

Transformation,  periodic,  93,  95. 

Transformation,  reciprocal  radii,  38,  41 ; 
on  the  sphere,  51. 

Transformation,  similarity,  32,  35. 

Transformations,  group  of,  90,  113. 

Translation,  parallel,  28,  30,  33. 

Transmission  of  electricity,  187;  of 
heat,  187. 

Triangles,  infinitesimal,  183  ;  similar,  136. 

Trigonometric  functions,  216-221 ;  addi 
tion  theorem  for,  218;  periodicity  of, 

222. 

Triple  numbers,  22. 

Unconditional  convergence,  170. 
Undetermined    coefficients,    method    of, 
207. 


Uniform  approach  to  a  limit,  132,  141, 
155- 

Uniform  convergence,  155,  156,  200,  202, 
203. 

Uniform  continuity,  132. 

" Uniformizing "  variable,  276. 

Unique  development  in  a  power  series, 
207. 

Unit  circle,  15,  40;  mapped  on  2-half- 
plane,  67  ;  reflection  on,  40. 

Units  for  complex  numbers,  8 ;  for  num 
ber-pairs,  7. 

Upper,  lower  bounds,  127,  128. 

Upper  bound  =  low  integral,  152. 

Upper  bound,  least,  128. 

Upper  integral,  152. 

Upper  limit  of  an  integral,  189. 

Value  of  a  function,  at  °o  and  =  oo, 
173;  in  a  domain,  196 ;  indeterminate, 
45,  103,  174. 

Values  of  a  periodic  function,  233. 

Variable,  definition  of  complex,  28;  uni- 
formizing,  276. 

Variation  of  a  quantity,  total,  175. 

VEBLEN  and  LENNES,  28,  131,  140,  162. 

Velocity  potential,  186,  187. 

WEBER,  317. 

WEIERSTRASS,   129,   168,   201,   228,   261, 

263,  382,  383  ;  expansion  in  a  product, 

373- 

WHITTAKER,  276. 
Winding-point,  395. 

YOUNG,  W.  H.  and  G.  C.,  136. 

Zero,  205,  211,  241;  at  oo ,  106;  corre 
sponds  to  oo ,  45,  104 ;  division  by,  45 ; 
many-fold,  102;  point,  102. 

Zeros  and  poles,  242  ;  at  a  branch-point, 
335;  number  of,  240-242. 


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